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NOV 9 1899 





Clacma^mt ||»8s §^mt» 









P. O. TAIT, M.A. 


warfdof dfydotf <pviX€OM fiii&ffuiT* ixowraaf. 


[The righti af tirantlaUon and reproduction are reserved,] 

Moih lUiXi. 

/0,J.V-cVu^''f S'' c<^. 




The present work was commenced in 1859^ while I was a 
Professor of Mathematics^ and far more ready at Quaternion 
analysis than I can now pretend to be. Had it been then 
completed I should have had means of testing its teaching 
capabilities^ and of improving it^ before publication^ where 
found deficient in that respect. 

The duties of another Chair^ and Sir W. Hamilton's wish that 
my volume should not appear till after the publication of his 
Elements J interrupted my already extensive preparations. I had 
worked out nearly all the examples of Analytical Geometry in 
Todhimter's Collection^ and I had made various physical appli- 
cations of the Calculus^ especially to Crystallography^ to Geo- 
metrical Optics, and to the Induction of Currents, in addition 
to those on Kinematics, Electrodynamics, &c., which are re- 
printed in the present work from the Quarterly Mathematical 
Jov/mal and the Proceedmgs of the Royal Society of Edinburgh. 

Sir W. Hamilton, a few days before his death, urged me to 
prepare my work as soon as possible, his being almost ready for 
publication. He then expressed, more strongly perhaps than 
he had ever done before, his profound conviction of the impor- 
tance of Quaternions to the progress of physical science ; and 


his desire that a really elementary treatise on the subject should 
soon be published. 

I regret that I have so imperfectly fulfilled this last request of 
my revered Mend. When it was made I was already engaged, 
along with Sir W. Thomson, in the laborious work of preparing 
a large Treatise on Natural Philosophy. The present volume 
has thus been written under very disadvantageous circumstances, 
especially as I have not found time to work up the mass of 
materials which I had originally collected for it, but which I 
had not put into a fit state for publication. I hope, however, 
that I have to some extent succeeded in producing a thoroughly 
elementary work> intelligible to any ordinary student ; and that 
the numerous examples I have given, though not specially 
chosen so as to display the full merits of Quaternions, will yet 
sufficiently show their admirable simplicity and naturalness to 
induce the reader to attack the Lectures and the Elements; 
where he will find, in provision, stores of valuable results, and 
of elegant yet powerM analytical investigations, such as are 
contained in the writings of but a very few of the greatest 
mathematicians. For a succinct account of the steps by which 
Hamilton was led to the invention of Quaternions, and for other 
interesting information regarding that remarkable genius, I 
may refer to a slight sketch of his life and works in the North 
British Review for September 1866. 

It will be found that I have not servilely followed even so 
great a master, although dealing with a subject which is en- 
tirely his own. I cannot, of course, tell in every case what I 
have gathered from his published papers, or from his voluminous 
correspondence, and what I may have made out for myself. 
Some theorems and processes which I have given, though wholly 
my own, in the sense of having been made out for myself before 
the publication of the Memetits, I have since found there. Others 


also may be, for I have not yet read that tremendous volume 
completely, since much of it bears on developments unconnected 
with Hiysics. But I have endeavoured throughout to point out 
to the reader all the more important parts of the work which 
I know to be wholly due to Hamilton. A great part, indeed, 
may be said to be obvious to any one who has mastered the 
preliminaries j still I think that, in the two last Chapters 
especially, a good deal of orig^al matter will be found. 

The volume is essentially a working one, and, especially in 
the later Chapters, is rather a collection of examples than a 
detailed treatise on a mathematical method. I have constantly 
aimed at avoiding too great extension; and in pursuance of 
this object have omitted many valuable elementary portions 
of the subject. One of these, the treatment of Quaternion 
logarithms and exponentials, I greatly regret not having given. 
But if I had printed all that seemed to me of use or interest ^ 
to the student, I might easily have rivalled the bulk of one of 
Hamilton's volumes. The beginner is reconmiended merely to 
read the first five Chapters, then to work at Chapters VI, VII, 
VIII (to which numerous easy Examples are appended). After 
this he may work at the first five, with their (more difficult) 
Examples ; and the remainder of the book should then present 
no difficulty. 

Keeping always in view, as the great end of every mathe- 
matical method, the physical applications, I have endeavoured to 
treat the subject as much as possible from a geometrical instead 
of an analytical point of view. Of course, if we premise the 
properties of i,/, k merely, it is possible to construct from them 
the whole system^; just as we deal with the imaginary of 

* This has been done by Hamilton himself, as one among many methods he 
has employed ; and it is also the foundation of a memoir by M. All^gret entitled 
Eisai 8ur le Galcul des Qttatemions (Paris, 1863). 


Algebra, or, to take a closer analogy, just as Hamilton himself 
dealt with Couples, Triads, and Sets. This may be interesting 
to the pure analyst, but it is repulsive to the physical student, 
who should be led to look upon i,y, h from the very first as 
geometric realities, not as algebraic imaginaries. 

The most striking peculiarity of the Calculus is that muliipli- 
cation is not generally commutative^ i. e. that qr is in general 
different from rq, r and q being quaternions. StiU it is to be 
remarked that something similar is true, in the ordinary coor- 
dinate methods, of operators and functions : and therefore the 
student is not wholly unprepared to meet it. No one is puzzled 
by the fact that log. cos. ^ is not equal to cos.log.a?, or that ^/j- 

is not equal to ^ Vy. Sometimes, indeed, this rule is most 
absurdly violated, for it is usual to take cos' a? as equal to (cos a?)», « 
while cos"*^ is not equal to (cosa?)"'^ No such incongruities 
appear in Quaternions ; but what is true of operators and 
frmctions in other methods, that they are not generally com- 
mutative, is in Quaternions true in the multiplication of (vector) 

It will be observed by those who are acquainted with the 
Calculus that I have, in many cases, not given the shortest or 
simplest proof of an important proposition. This has been done 
with the view of including, in moderate compass, as great a 
variety of methods as possible. With the same object I have 
endeavoured to supply, by means of the Examples appended to 
each Chapter, hints (which will not be lost to the intelligent 
student) of further developments of the Calculus. Many of 
these are due to Hamilton, who, in spite of his great originality, 
was one of the most excellent examiners any University can 
boast of. 

It must dways be remembered that Cartesian methods are 


mere particular cases of Quaternions, where most of the dis- 
tinctive features have disappeared; and that when, in the treat- 
ment of any particular question, scalars have to be adopted^ the 
Quaternion solution becomes identical with the Cartesian one. 
Nothing therefore is ever lost, though much is generally gained, 
by employing Quaternions in preference to ordinary methods. 
In fact, even when Quaternions degrade to scalars, they give 
the solution of the most general statement of the problem they 
are applied to, quite independent of any limitations as to choice 
of particular coordinate axes. 

There is one very desirable object which such a work as 
this may possibly fulfil. The University of Cambridge, while 
seeking to supply a real want (the deficiency of subjects of 
examination for mathematical honours, and the consequent fre- 
quent introduction of the wildest extravagance in the shape of 
data for " Problems^'), is in danger of making too much of such 
elegant trifles as Trilinear Coordinates, while gigantic systems 
like Invariants (which, by the way, are as easily introduced into 
Quaternions as into Cartesian methods) are quite beyond the 
amount of mathematics which even the best students can master 
in three years' reading. One grand step to the supply of this 
want is, of course, the introduction into the scheme of examina- 
tion of such branches of mathematical physics as the Theories of 
Heat and Electricity. But it appears to me that the study of a 
mathematical method like Quaternions, which, while of immense 
power and comprehensiveness, is of extraordinary simplicity, and 
yet requires constant thought in its applications, would also be 
of great benefit. With it there can be no " shut your eyes, and 
write down your equations/' for mere mechanical dexterity of 
analysis is certain to lead at once to error on account of the 
novelty of the processes employed. 

The Table of Contents has been drawn up so as to give the 



student a short and simple smnmary of the chief fundamental 
formulae of the Calculus itself^ and is therefore confined to an 
analysis of the first five chapters. 

In conclusion^ I have only to say that I shall be much obliged 
to any one^ student or teacher^ who will point out portions of 
the work where a difficulty has been found ; along with any 
inaccuracies which maybe detected. As I have had no assistance 
in the revision of the proof-sheets^ and have composed the work 
at irr^;ular intervals^ and while otherwise laboriously occupied^ 
I fear it may contain many slips and even errors. Should it 
reach another edition there is no doubt that it will be improved 
in many important particulars. 

P. G. TAIT. 

Jvly 1867. 


Chapter I. — ^Vectors and their Composition 1-29 

Sketch of the attempts made to represent geometrically the imaginary of 

algebra, §§ 1-13. 
De Moivre's Theorem interpreted in plane rotation, § 8. Gniious speoala- 

tion of Servois, § 11. 
Elementary geometrical ideas connected with relative position, § 15. De- 
finition of a Veotob. It may be employed to denote transloHan, 

Expression of a vector by one symbol, containing implicitly three distinct 

numbers. Extension of the signification of the symbol «, § 18. 
The sign + defined in accordance with the interpretation of a vector as 

representing translation, § 19. 
Definition of — . It simply reverses a vector, § 20. 
Triangles and polygons of vectors, analogous to those of forces and of 

simnltaneons velocities, § 21. 
When two vectors are paraUd we have 
' a»2^, § 22. 

Any vector whatever may be expressed in terms of three distinct vectors, 
which are not coplanar, by the formula 

p^xa + yfi + zy, 
which exhibits the three numbers on which the vector depends, § 28. 
Any vector in the same plane with a and fi may be written 

p^xa + yfi, § 24. 
The equation •■=Pf 

between two vectors, is equivalent to three distinct equations among 
numbers, § 25. 


The Commutative and Associative Laws hold in the combination of vectors 

by the signs + and — , § 27. 
The equation p = x$, 

where p is a variable, and fi a fixed, vector, represents a line drawn 

through the origin parallel to 0. 
p=^a + x0 

is the equation of a line drawn through the extremity of a and parallel 

to /B, § 28. 

p = ya + x$ 

represents the plane through the origin parallel to a and fi, § 29. 
The condition that p, a, fi may terminate in the same line is 
pp + qa + rfi = 0, 

subject to the identical relation 

p + q + r = 0. 
Similarly pp + qa + rfi + 8y==0, 

with, p + q + r + 8 = 0y 

is the condition that the extremities of four vectors lie in one plane, 

§ 30. 
Examples with solutions, § 31. 
Differentiation of a vector, when given as a function of one number, 

§§ 32-38. 
If the equation of a curve be 

P = it>{8) 

where a is the length of the arc, dp is a vector tangent to the curve, 
and its length is ds, §§ 38, 39. 
Examples with solutions, §§ 40-44. 

Examples to Chapter 1 29-81 

Chapter II. — Products and Quotients of Vectors ... 32-59 

Here we begin to see what a quaternion is. When two vectors are 
parallel their quotient is a number, §§ 45, 46. 

When they are not parallel the quotient in general involves faur distinct 
numbers — and is thus a Quaternion, § 47. 

A quaternion regarded as the operator which turns onft vector into 
another. It is thus decomposable into two factors,' whose order is 
indifferent, the stretching factor or Tensor, and the turning factor or 
Versor. These are denoted by Tq and Uq, § 48. 

The equation fi = qa 

gives — — 3» or i8<»~^ = g'» hut not in general 

a-^fi^q, § 49. 
q or $a-^ depends only on the relative lengths, and directions, of fi and a, 


Reciprocal of a quaternion defined, 

5 = - gives- or q » = -, 

T.?-^=^-, ILq-'^-^Uq, §51. 

Definition of the Conjiigate of a quaternion, 
and qKq = Kq.q = {Tq)\ §52. 
Representation of yersors by arcs on the unit-sphere, § 53. 
Versor multiplication illustrated by the composition of arcs, § 54. 
Proof that K{qr) = Kr.Kq, §55. 

Proof of the Associative Law of Multiplication 
p.qr=^pq.r, §§ 57-60. 
[Digression on Spherical Conies, § 59*.] 
Quaternion addition and subtraction are commutative, § 61. 
Quaternion multiplication and division are distribtUivet § 62. 
Composition of quadrantal versors in planes at right angles to each other. 
Calling them t, j, k, we have 
i»=ja=&«=_l. ij^-^ji=h, jk=-ij=i, ki=^-ik^j, 

ijk=-l, §§ 64-71. 
A unit-vector, when employed as a factor, may be considered as a qua- 
drantal versor whose plane ia perpendicular to the vector. Hence the 
equations just written are true of any set of rectangular unit-vectors 
i,j,k, §72. 
The product, and quotient, of two vectors at right angles to each other is 
a third perpendicular to both. Hence 

and (T'a)«=aKi«=-o», § 78. 
Eveiy versor may be expressed as a power of some unit- vector, § 74. 
Every quaternion may be expressed as a power of a vector, § 76. 
The Index Law is true of quaternion multiplication and division, § 76. 
Quaternion considered as the sum of a Soalab and Yectob. 

q::^-^x + y^Sq+Vq, §77. 

Proof that SKq^^Sq, VKq=^ - Vq, § 79. 

Quadrinomial expression for a quaternion 

q=^w + ix+jy + kz. 

An equation between quaternions is equivalent to four equations between 

numbers (or scalars), § 80. 
Second proof of the distributive law of multiplication, § 81. 
Algebraic determination of the constituents of the product and quotient of 

two vectors, §§ 82-84. 


Heewj^ prryiif o^ the nM9fi0'yttiz*. Uw of mTiItli>L:«:aar»n, § *5. 
PrrjT/f of the 6>nQ'ilAe 

Proof of the iomnilae 

Fu^^- F.7a«, 

= Faa9ya+ F^&i5+ F7«5A §$ 90.92. 
Hamilton'f proof thai the product of two pMallel Tecton must be a scalar, 
and thai of perpendicalar vecton, a vector ; if qvatanions are to deal 
with space indifferently in all directions, § 93. 

Examples to Chapter II 59, 60 

Chapteb ni. — Intekpeetations and Transpormattons op 


If be the angle between two yectors, a and 0, we have 
5^- = ^^c08«, Sa0=^TaT$cos$, 

TV-^^sm0, TVafi=TaTfiam$. 

a la 

Applications to plane trigonometiy, §§ 94-97. 
shows that a is perpendicular to /3, while 

shows that a and $ are paralleL 

is the Yolmne of the parallelepiped three of whose contenninouB edges 
are a, fi,y. Hence 

shows that a, $,y are coplanar. 
Expression of S.affy as a determinant, §§ 98-102. 
Proof that (TqY^iSqy^ + {TVq)\ 

and T(2r) = rgrr, §108. 

Simple propositions in plane trigonometry, § 104. 

Proof that ofiaT^ is the vector reflected ray, when /9 is the incident ray 
and a normal to the reflecting surface, § 105. 


Interpretation of oiB^ when it is a vector, § 106. ' 

Examples of variety in simple transformations, § 107. 

Introduction to spherical trigonometry, §§ 108-113. 

Representation, graphic, and by quaternions, of the spherical eicess, 

§§ 114, 116. 
Lod represented by different equations — ^points, lines, surfaces, and solids, 

§§ 116-119. 
Proof that ^ 

r-»(r«2^*5-i«i7(r2 + J&iC2). § 120. 
Proof of the transformation 

BiQUATESNIONS, §§ 123-125. 

Conyenient abbreviations of notation, §§ 126, 127. 

£cXAMFLfls TO Chafteb III 85-8S 

Chaptbe rV. — Differentiation of Quatebnions. . . 89-96 

Definition of a differential, 

dr^dFi^^n j i? (j + ^-) -Fq\, 

where dq is any quaternion whatever. 
We may write 


where / is linear and homogeneous in dq ; but we cannot generally 

write dFq=f(q)dq, §§128-131. 

Definition of the differential of a function of more quaternions than one. 

d (qr) =s qdr + dq.r, but not generaUy d (qr) «» qdr + rdq. § 132. 
Proof that dTp ^dp 

Up p 

Successive differentiation; Taylor's theorem, §§ 134, 135. 

If the equation of a surface be 

the differential may be written 
where v is a vector normal to the sur&ce, § 137. 
ExAHPLBS TO Chapter IY 96 


Chaptee V. — ^Thb Solution of Equations of the Fibst Degree. 


The most general equation of the first degree in an unknown quaternion q, 
may be written 

^VMqh-\ 8.cq = d, 
where a, h, c, d 9ie given quaternions. Elimination of Sq, and reduc- 
% tion to the vector equation 

^p = 2.oSi3p = 7, §§ 138, 139. 
General proof that ^'p is expressible as a linear function oi p, <pp, and ^'''p, 

§ 140. 
Value of ^ for an ellipsoid, employed to illustrate the general case, 

§§ 141-143. 
Hamiltop's solution of 

If we write Sa^ = Sp^'tr^ 

the functions ^ and ^' are said to be conjugate, and 

Proof that m, whose value may be written as 

is the same for all values of A, ijl^v, §§ 144-146. 
Proof that if mg^m-\rm^g-\-m^g^ +g^, 


S {\fJUt>'v + p'XfJLP + A^ V) 
S.Kfiv * 

then mg{<p + g)~^ V\fi = (m<f>~^ + g'X + </'*) ^^M- 
Also that X = »*a — ^> 

whence the final form of solution 

m<p-^ =:mi -m,^ + ^% §§ 147, 148. 
Examples, §§ 149-161. 
The fundamental cubic 

ip'^-m^<t>^ + m^il>-m'='{<p-g^){<f,-g^){<p-g^) = 0. 
When ^ is its own conjugate, the roots of the cubic are real; and the 
equation yp^P = ^t 

or {<t>-g)p=0, 
is satisfied by a set of three real and mutually perpendicular vectors. 
Geometrical interpretation of these results, §§ 162-166. ^ 

Proof of the transformation 

<t>p ^pp + q V.i% + €k)p{l — ck) 
where (<^ — </i)* = 0, 

{<l>-g,)k = 0, 


Another transformation is 

il>pm,aaVap + h$S$p, §§ 167-169. 
Other properties of ^. Proof that 

Sp(!p+9)''^P^0, and Spdp + h)'^ p^'^O 
represent the same snrfiice if 

Proof that when ^ is not self-conjugate 
Proof that, if g^o^ + iS^/S + T^, 

where a, iS, 7 are any rectangular unit vectors whatever, we have 
Sq^-m^, Fg«€. §§170-174. 

Degrees of indeterminateness of the solution of a quaternion equation — 

Examples, §§ 175-179. 
The linear function of a quaternion is given by a symbolical biquadratic, 

Particular forms of linear equations, §§ 181-183. 
A quaternion equation of the mth degree in general involves a scalar 

equation of degree mS § 184. 
Solution of the equation 

q*^qa + h, § 185. 
EXAMFLBS TO Chafteb V 128-130 

Chaptbk VI. — Oeometbt op the Straight Line and Plane. 


ExAMPT.iBH TO Chaiteb YL 148-150 

Chapter VII. — ^Thb Sphere and Cyclic Cone. . . 151-167 

EXAUFLBS TO Chapteb YII 167-170 

Chapter VIII. — Surfaces op the Second Order. . 171-191 

Examples to Chaftib Yin 192-ld6 



Chapter IX. — Geometry of Curves and Surfaces. 197-238 
Examples to Chaftbr IX 238-247 

Chapter X. — Kinematics . , . 248-271 

Examples to Chapter X 271-275 

Chapter XI. — ^Physical Applications 276-311 

Miscellaneous Examples 312-320 


Page 171, first line of § 249, fw OD read OD. 
,, 213, last line but one, for Sctfip read S.afip. 
,, 225, Une 10, /or equations read equation. 





^OR more than a century and a half the geometrical 
representation of the negative and imaginary alge- 
braic quantities,— 1 and \/— 1, or, as some prefer to write them, 
— and — ^, has been a favourite subject of speculation with 
mathematicians. The essence of almost all of the proposed 
processes consists in employing such quantities to indibate the 
direction, not the length, of lines. 

2. Thus it was soon seen that if positive quantities were 
measured off in one direction along a fixed line, a useful and 
lawful convention enabled us to express negative quantities by 
simply laying them off on the same line in the opposite direction. 
This convention is an essential part of the Cartesian method, 
and is constantly employed in Analytical Geometry and Applied 

3. Wallis, in the end of the seventeenth century, proposed to 
represent the impossible roots of a quadratic equation by going 
out of the line on. which, if real, they would have been laid off. 

\ B 


His construction is equivalent to the consideration of \/^— 1 
as a directed unit-line perpendicular to that on which real 
quantities are measured. 

4. In the usual notation of Analytical Geometry of two 
dimensions, when rectangular axes are employed^ this amounts 
to reckoning each unit of length along 0^ as -h a/— T, and on 
0^ as --\/— 1 ; while on Ox each unit is -f 1, and on Ox' it 
is — 1 . K we look at these four lines in circular order, i. e. in 
the order of positive rotation (opposite to that of the hands 
of a watch), they give 

In this series each expression is derived from that which pre- 
cedes it by multiplication by the factor \/— 1. Hence we may 
consider \/— 1 as an operator, analogous to a handle perpen- 
dicular to the plane of xyy whose eflPect on any line is to make 
it rotate (positively) about the origin through an angle of 90°. 

5. In such a system, a point is defined by a single imaginary 
expression. Thus a'\-h>J—\ may be considered as a single 
quantity, denoting the point whose coordinates are a and h. 
Or, it may be used as an expression for the line joining that 
point with the origin. In the latter sense, the expression 
a-\-hs/ —X implicitly contains the direction^ as well as the lengthy 
of this line ; since, as we see at once, the direction is inclined 

at an angle tan~* - to the axis of a?, and the length is V^a' + i*. 


6. Operating on this symbol by the factor \/— 1, it becomes 
— i -h dj \/— 1 ; and now, of course, denotes the point whose x and 
y coordinates are —J and a; oi* the line joining this point with 
the origin. The length is still \/fl* -f i*, but the angle the line 

makes with the axis of x is tan** ( ~ r) ^ ^^^^^ ^ evidently 90° 
greater than before the operation. 


7. De Moivre's Theorem tends to lead us still farther in 
the same direction. In feet, it is easy to see that if we use, 
instead of >>/— 1, the more general factor cosa+ \/ — 1 sin a, 
its effect on any line is to turn it through the (positive) angle a 

in the plane of of, y, [Of course the former factor, \/— 1, is 

merely the particular case of this, when a = - •] 

Thus (cosa-f a/— 1 sina)(«-f i>/— 1) 

= Acosa— 3sina+ \/ — 1 (a sin a + i cos a), 
by direct multiplication. The reader will at once see that the 
new form indicates that a rotation through an angle a has taken 
place, if he compares it with the common formulae for turning 
the coordinate axes through a given angle. Or, in a less simple 
manner, thus — 

Length = V'(«cosa— Jsina)*-f (asina + Jcosa)' 
= A/^'-fd* as before. 

Inclination to axis of or ^ 

, tan a + - 

, Asma + ^cosa ^ , a 

= tan-* r-^ — = tan" 

«coso— isina , h, 

1 tana 

h ^ 

= a-f tan-*-- 

8. We see now, as it were, why it happens that 

(cos a4- a/— 1 sin a)** = cos ma-f >/— 1 sin ma. 
In fact, the first operator produces m successive rotations in the 
same direction, each through the angle fi; the second, a single 
rotation through the angle ma. 

9. It may be interesting, at this stage, to anticipate so far as 
to state that a Quaternion can, in general, be put under the form 

where iV is a numerical quantity, Q a real angle, and 

tsr»= — 1. 
B 2 


This expression for a quaternion bears a very close analogy to 
the forms employed in De Moivre's Theorem ; but there is the 
essential difference (to which Hamilton's chief invention referred) 
that cr is not the algebraic \/— 1, but may be any directed unit* 
line whatever in space. 

10. In the present century Argand, Warren, and others, ex- 
tended the results of Wallis and De Moivre. They attempted 
to express as a line the product of two lines each represented by 
a symbol such as a + iv^— 1. To. a certain extent they suc- 
ceeded, but simplicity was not gained by their methods, as the 
terrible array of radicals in Warren's Treatise sufficiently proves. 

IL A very curious speculation, due to Servois, and published 
in 1813 in Gergonne's Annates y is the only one, so far as has 
been discovered, in which the slightest trace of an anticipation of 
Quaternions is contained. Endeavouring to extend to space the 
form «-f i^/— 1 for the plane, he is glided by analogy to write 
for a directed unit-line in space the form 

p cos a-l-5' cos ^ -f r cos y, 

where a, j3, y are its inclinations to the three axes. He perceives 
easily that p, q, r must be non-reals : but, he asks, ^^ seraient- 
elles imaginaires reductibles k la forme generale ^-f jBa/— 1 ?'' 
This he could not answer. In feet they are the i,j, h of the 
Quaternion Calculus. (See Chap. II.) 

12. Beyond this, few attempts were made, or at least re- 
corded, in earlier times, to extend the principle to space of thre^ 
dimensions ; and, though many such have been made within the 
last forty years, none, with the single exception of Hamilton's, 
have resulted in simple, practical methods; all, however in- 
genious, seeming to lead at once to processes and results of 
fearful complexity. 

For a lucid, complete, and most impartial statement of the 


claims of his predecessors in this field we refer to the Preface to 
Hamilton's Lectures on Quaternions. 

13. It was reserved for Hamilton to discover the use of >/— 1 
as a geometric realUy, tied down to no particular direction in 
space, and this use was the foundation of the singularly elegant, 
yet enormously powerful. Calculus of Quaternions. 

"While all other schemes for using \/— 1 to indicate direc- 
tion make one direction in space expressible by real numbers, 
the remainder being imaginaries of some kind, leading in general 
to equations which are heterogeneous ; Hamilton makes all di- 
rections in space equally imaginary, or rather equally real, 
thereby ensuring to his Calculus the power of dealing with 
epace indiflferently in all directions. 

In fact, as we shall see, the Quaternion method is independent 
of axes or any supposed directions in space, and takes its refer- 
«nce lines solely from the problem it is applied to. 

14. But, for the purpose of elementary exposition, it is best 
to begin by assimilating it as closely as we can to the ordinary 
Cartesian methods of Geometry of Three Dimensions, which are 
in fact a mere particular case of Quaternions in which most 
of the distinctive features are lost. We shall find in a little 
that it is capable of soaring above these entirely, after having 
employed them in its establishment; and, indeed, as the in- 
ventor's works amply prove, it can be established, ab initiOy in 
various ways, without even an allusion to Cartesian Geometry. 
As this work is written for students acquainted with at least 
the elements of the Cartesian method, we keep to the first- 
mentioned course of exposition; especially as we thereby avoid 
some reasoning which, though rigorous and beautiftil, might be 
apt, fix)m its subtlety, to prove repulsive to the beginner. 

We commence, therefore, with some very elementary geo- 
metrical ideas. 


15. Suppose we have two points A and B in, ipace^ and sup- 
pose A given^ on how many numbers does W^ relative position 
depend ? 

If we refer to Cartesian coordinates (rectangular or not) we 
find that the data required are the excesses of -B's three coor- 
dinates over those of A, Hence three numbers are required. 

Or we may take polar coordinates. To define the moon's 
position with respect to the earth we must have its Geocentric 
Latitude and Longitude^ or its Bight Ascension and Declination, 
and^ in addition^ its distance or radius- vector. Three again. 

16. Here it is to be carefiiUy noticed that nothing has been 
said of the actual coordinates of either A or B^ or of the earth 
and moon^ in space ; it is only the relative coordinates that are 

Hence any expression^ as AJBy denoting a line considered with 
reference to direction as well as length, contains implicitly three 
numbers, and all lines parallel and equal to AB depend in the 
same way upon the same three. Hence, all lines which are 
equal and parallel may be represented hy a common symioly and 
that symbol contains three distinct numbers. In this sense a Une 
is called a vbctoe, since by it we pass from the one extremity, 
A, to the other, B] and it may thus be considered as an in- 
strument which carries A to Bi so that a vector may be em- 
ployed to indicate a definite translation in space. 

17. We may here remark, once for all, that in establishing 
a new Calculus, we are at liberty to give any definitions what- 
ever of our symbols, provided that no two of these interfere 
with, or contradict, each other, and in doing so in Quaternions 
simplicity and (so to speak) naturalness were the inventor's aim. 

18. Let AB be represented by a, we know that a depends 
on three separate numbers. Now if CD be equal in length to AB 


and if these lines be parallel^ we have evidently CD = AB = a, 
where it will be seen that the sign of equality between vectors 
contains implicitly equality in length and parallelism in direction. 
So far we have extended the meaning of an algebraic symbol. 
And it is to be noticed that an equation between vectors, as 

o = /3, 

contains three distinct equations between mere numbers. 

19. We must now define -f (and the meaning of — will 
follow) in the new Calculus. Let Ay By C be any three points 
and (with the above meaning of =) let 

lB = ay BC=:py IC=zy. 

If we define + (in accordance with the idea (§ 16) that a 
vector represents a translation) by the equation 

or iB^BC^ACy 

we contradict nothing that precedes, but we at once introduce 
the idea that vectors are to be compoundedy in direction and 
magnitudey like simultaneous velocities, A reason for this may 
be seen in another way if we remember that by adding the 
differences of the Cartesian coordinates of A and By to those of 
the coordinates of B and C, we get those of the coordinates of 
A and C. 

20. But we also see that if C and A coincide (and C may be 
fl^^y point) AC^Qy 

for no vector is then required to carry A to C. Hence the above 
relation may be written, in this case, 

or, introducing, and by the same act defining, the symbol — , 
52= -iB. 


Hence, the symbol — , applied to a vector , simply shows that its 
direction' is to be reversed. 

And this is consistent with all that precedes ; for instance, 

IB + BC= AC, 
and A£ = IC-BC, 

or -IC+CB, 

are evidently but different expressions of the same truth. 

2L In any triangle, ABC, we have, of course, 

AB + BC+CA = 0; 
and, in any closed polygon, whether plane or gauche, 

AB+BC+ -^YZ^ZA^O. 

In the case of the polygon we have also 

ZB-f J^-f J^fZ^M 

These are the well-known propositions regarding composition 
of velocities, which, by the second law of motion, give u» the 
geometrical laws of composition of forces. 

22. If we compound any number of parallel vectors, the 
result is obviously a numerical multiple of any one of tbem^ 

Thus, if A, B, C are in one straight line, 


where <r is a number, positive when B lies between A and C, 
otherwise negative : but such that its numerical value, inde- 
pendent of sign, is the ratio of the length of BC to that of AB. 
This is at once evident if AB and BC be commensurable ; and 
is easily extended to incommensurables by the usual reductio ad 

23. An important, but almost ob^dous, proposition is that 
any vector may he resolved into three components parallel respectively 


to any three given vectors ^ no two of which are parallel, and which 

are not parallel to one plane. 

Let OA, OB, OC be the three fixed 
vectors, OP any other vector. From 
P draw PQ parallel to CO, meeting the 
plane BOA in Q. [There mast be a 
real point Q, else PQ, and therefore 
CO, would be parallel to BOA, a case 
specially excepted.] From Q draw QR 
parallel to BO, meeting OA in R, Then 

we have 6P=6R + RQ-^QP (§ 21), 

and these components are respectively parallel to the three given 
vectors. By § 22 we may express OR as a numerical multiple 
of OA, RQ o{ OB, and QP of OC, Hence we have, generally, 
for any vector in terms of three fixed non-coplanar vectors, a, ^, y, 

"which exhibits, in one form, the three numbers on which a 
vector depends (§ 16). Here x,y, z are perfectly definite. 

24. Similarly any vector, as OQ, in the same plane with OA 
and OB, can be resolved into components OR, RQ, parallel 
respectively to OA and OB; so long, at least, as these two 
vectors are not parallel to each other. 

25. There is particular advantage, in certain cases, in employ •. 
ing a series of three mutually perpendicular unit-vectors as lines 
of reference. This system Hamilton denotes by i,j, h. 

Any other vector is then expressible as 
p = xi'\-yj-^zk. 
Since iyj, k are unit- vectors, x, y, z are here the lengths of con- 
terminous edges of a rectangular parallelepiped of which p is the 
vector-diagonal ; so that the length of p is, in this case, 



Let or = fi-f ij^ + f* 

be any other vector, then the vector equation 

p ^ V 
obviously involves the following three equations among numbers^ 

^ = f , y = n, ^ = f • 
Suppose i to be drawn eastwards, J northwards, and i upwards, 
this is equivalent merely to saying that if two points coincide, 
they are equally to the east {or west) of any third pointy equally 
to the north {or south) of it, and equally elevated above {or depressed 
below) its level. 

26. It is to be carefiilly noticed that it is only when o, p, y 
are not eoplanar that a vector equation such as 

p = Iff, 

or iPa-fy/3+;2ry = fa-f r/^-f fy, 

necessitates the three numerical equations 

For, if o, )8, y be eoplanar (§ 24), a condition of the following form 

must hold 1 a 

y = aa-\-op. 

Hence p = (x'\'Za)a-\-{]/-\-zb)fi, 

^= (f + i'«)a + ('7 + f«)/3, 

and the equation p ■= vr 

now requires only the two numerical conditions 

x-\-za:=: (+Ca, y-\-zb = 17+f*. 

27. The Commutative and Associative Laws hold in the com-- 
bination of vectors by the signs + and — . It is obvious that, if 
we prove this for the sig^ -f , it will be equally proved for — , 
because — before a vector ($ 20) merely indicates that it is to 
be reversed before being considered positive. 

Let A, By Cy B be, in order, the comers of a parallelogram ; 
we have, obviously. 


And AB^SC^ IC= AD-\-BC ^ BC^lB. 

TIence the commutatiye law is true for the addition of any two 

vectors, and is therefore generally true. 

Again, whatever four points are represented by A, B, C, B, 
we have 

AB = AB-^BB = IC+ CB, or 

ZB-f JC'+CS = AB-\-{BC+CB) = {AB^BC)+CB. 
And thus the truth of the associative law is evident. 

28. The equation ^ 

/> = ^P> 

where p is the vector connecting a variable point with the origin, 
P a definite vector, and x an indefinite number, represents the 
straight line drawn from the origin parallel to /3 (§ 22). 

The straight line drawn from J, where OA = a, and parallel 
to j3, has the equation 

p = a+^j3 (1) 

In words, we may pass directly from to P by the vector OP 
OT p; or we may pass first to A, by means of OA or o, and then 
to P along a vector parallel to )3 (§ 16). 

Equation (1) is one of the many usefiil forms into which 
Quaternions enable us to throw the general equation of a straight 
line in space. As we have seen (§ 25) it is equivalent to tAree 
numerical equations ; but, as these involve the indefinite quan- 
tity a?, they are virtually equivalent to but two, as in ordinary 
Geometry of Three Dimensions. 

29. A good illustration of this remark is furnished by the 

fact that the equation 

p =ya-f-a?/3, 

which contains two indefinite quantities, is virtually equivalent 
to only one numerical equation. And it is easy to see that 
it represents the plane in which the lines a and fi lie ; or the 

C 2 • 

12 QUATERNIONS. [chap. I. 

surface which is formed by drawing, through every point of OA, 
a line parallel to OJB. In fact, the equation, as written, is simply 
§ 24 in symbols. 

And it is evident that the equation 

is the equation of a plane passing through the extremity of y, 
and parallel to a and j3. 

It will now be obvious to the reader that the equation 

P =i?iai+J»aa2+ = 2i?a, 

where Oi, aa, &c. are given vectors, and jOi, j»2, &c. numerical 
quantities, represe^its a straight line if jOj, p^y &c. be linear 
functions of one indeterminate number ; and a plane, if they be 
linear expressions containing two indeterminate numbers. Later 
(§31 (/)), this theorem will be much extended. 

30. The equation of the line joining any two points A and B^ 
where 0A= a and OB = j8, is obviously 

p = a + ^(i3 — a), 
or /) =^-fy(a-/3). 

These equations are of course identical, as may be seen by 
putting 1 — y for x. 

The first may be written 

p + {X'-l)a—xp = 0; 
or i?p + ^a + r/3 = 0, 

subject to the condition p-{-q-\-r = identically. That is — 
A homogeneous linear function of three vectors, equated to zero, 
expresses that the extremities of these vectors are in one straight 
line, if the sum of the coefficients be identically zero. 

Similarly, the equation of the plane containing the extremities 
A) By C of the three non-coplanar vectors a, j8, y is 

p = a+ir(/3-a)4-j^(y-^), 
where x and y are each indeterminate. 


This may be written 

^ with the identical relation 

p-^q + r-hs=0, 
which is the condition that four points may lie in one plane. 

3L We have already the means of proving, in a very simple 
manner, numerous classes of propositions in plane and solid 
geometry. A very few examples, however, must suffice at this 
stage; since we have hardly, as yet, crossed the threshold of 
the subject, and are dealing with mere linear equations con- 
necting two or more vectors, and even with them we are re- 
stricted as yet to operations of mere addition, 

(a^ The diagonals of a parallelogram bisect each other. 

Let ABCB be the parallelogram, the point of intersection 
of its diagonals. Then 

which gives 


The two vectors here equated are parallel to the diagonals re- 
spectively. Such an equation is, of course, absurd unless 

(1) The diagonals are parallel, in which case the figure 

is not a parallelogram ; 

(2) Id = 6C, and BO =: OB, the proposition. 

(i.) To show that a triangle can he constructed, whose sides 
are parallel, and equal, to the bisectors of the sides of 
any triangle. 

Let ABC be the triangle, Aa, Bb, Cc the bisectors of the 
sides. Then 

: A 


5« . . - =Sc+ic2, 

tfc . - - =CA-^iA£. 

Hence ia+5i + ^= |(i5+^+C5) = 0; ;! * 

which (§21) proves the proposition. ] 

Also 2^ = ZB + i^ ' 

_ _ _ ^ 

= i(i5-(7T) = iilS-i-IC), 

results which are sometimes useful. They may be easily veri^ 
by producing Aa to twice its length and joining the extrei^ 
with £, t ' 

(i'.) Tke bisectors of the sides of a triangle meet in a poi$i^ 
which trisects each of them. 

Taking A as origin, and putting a, )8, y for vectors parallelj^ 
and equal, to the sides taken in order; the equation of Bb is 
(§ 28 (1)) 

p = y + J?(y + -|.) = (H.a;)y + -|-^. 

That of Cc isj in the same way, 

At the point 0, where Bb and Cc intersect, 

p = (l+a;)y + |^=-(H-y)^-|y. 
Since y and /3 are not parallel, this equation gives 
• l+ar = -|, and | = -(1+^). 

From these ar = y = — § . 

Hence iO = i (y-^S) = %Ia. (See Ex. {b).) 



This equation shows, being a vector one, that Aa passes through 
Oy and that A0\ Oa\\2\\. 

(c.) If 62= a, dB=z p, 6C= aa-^-b^, 

be three given co-planar vectors, and the lines indicated 
in the figure be drawn , the points a, )3, y lie in a 
straight line. 
We see, at once, by the process indicated in § 30, that 

Oc= ^ ^ 

Hence we easily find 


Oa = - 


Ofi = 


1 — 2a— i' 

Oy = 

— ga + 3/3 

1— (»- 
These give 

— («)0^ + (l-2«-.*)0^-(5-fl)Oy = 0. 
But —(l*-a—2*)-h(l—2«— *)—(*— a) = identically. 
This, by § 30, proves the proposition. 

(d.) Let OA = a, OJB = j3, be any two vectors. If MP be 
parallel to OB ;- and OQ, BQ, be drawn parallel to 
APy OP respectively ; the locus of Q is a straight line 
parallel to OA. 

Let OM = ea. 

Hence the equation 
of OQ is 

9 =^(^^a-hiP/3); 
and that of BQ is 

p = &^z{ea'\-xfi). 


At Q we have, therefore, 

Xy=: \-\-ZXy ^ 

These give xy -z^ e, and the equation of the locus of Q is 

P = ^^+/a, 
i. e. a straight line parallel to OA, drawn through N in OB 
produced, so that 

ON '.OB 11 OMiOA. 

CoE. If BQ meet MP inq,Pq = fi; and if JP meet NQ mp, 

Also, for the point R we have pR = AF, QR = Bq. 

Hence if from any two points, A and By lines be drawn intercept^ 
ing a given length Pq on a given line Mq ; and if, from R their 
point of intersection, Rp be laid off = PA, and RQ ^ qB ; Q and 
p lie on a fixed straight line, and the length of Qp is constant. 

(e.) To find the centre of inertia of any system. 

If OA = a, OB = ai, be the vector sides of any triangle, the 
vector from the vertex dividing the base AB in C so that 
BC : CA :: mim^ is 

For AB is ai — a, and therefore ^Cis 



Hence 00^61+ AC 



This expression shows how to find the centre of inertia of two 
masses ; m at the extremity of a, m,^ at that of aj. Introduce m^ 
at the extremity of a^, then the vector of the centre of inertia 
of the three is, by a second application of the formula. 

m + nii 



For any number of masses, expressed generally by m at the 
extremity of the vector a, we have the vector of the centre of 

inertia ' _/ \ 

_ 2 (ma) 

This may be written 

2m(a--/3) = 0. 

Now ai--)3 is the vector of m^ with respect to the centre of 
inertia. Hence the theorem, If the vector of each element of a 
mass, drawn from the centre of inertia, be increased in length in 
proportion to the mass of the element, the sum of all these vectors 
is zero, 

{f) We see at once 
that the equation 


where ^ is an indeter- 
minate number, and a, j3 
given vectors, represents 
a parabola. The origin, 
0, is a point on the 
curve, /3 is parallel to the 
axis, i. e. is the diameter 

OB drawn from the origin, and a is OA the tangent at the 

origin. In the figure 


The secant joining the points where t has the values t and l^ 
is represented by the equation 

p = at+^-^x{at'-^^-at-^) (§30) 

= at + ^ + x{€-(){a+p^^}. 

Put f::^tf and write x {or x{f'~() [which may have any value] 
and the equation of the tangent at the point {f) is 

rva x=—t, p = — , 

or the intercept of the tangent on the diameter is — the abscissa 
of the point of contact. 

Otherwise: the tangent is parallel to the vector a+^t or 

at-^pt^ or a^ + ^ + ^or OQ+OP. But?y=fO+QP, 
hence TO ^ OQ. 

{0.) Since the equation of any tangent to the parabola is 

let us find the tangents which can be drawn from a given point. 
Let the vector of the point be 

p=^a+?/3 (§24). 
Since the tangent is to pass through this point, we have, as 
conditions to determine t and x, 

t+x =jp, 

— -\^xt±zq; 

by equating respectively the coefficients of a and p. 
Hence ^ =jo + >/jo* — 25'. 


Thus, in general, two tangents can be drawn from a given 
point. These coincide if 

P' = 2q; 
that is, if the vector of the point from which they are to be 
drawn is p^ 

i. e. if the point lies on the parabola. They are imaginary if 
2 q >jt?'», i. e. if the point be 

r heing positive. Such a point is evidently within the curve, as 
at R, where OQ = ^/3, QP ^pa, PB = rp. 

Ih,) Calling the values of t for the two tangents found in 
(^) t^ and t^ respectively, it is obvious that the vector 
joining the points of contact is 

which is parallel to t +t 

or, by the values of t^ and t^ in (^), 

Its direction, therefore, does not depend on q. In words, If 
pairs of tangents be drawn to a parabola from points of a diameter 
produced y the chords of contact are parallel to the tangent at the 
vertex of the diameter. This is also proved by a former result, for 
we must have OT ior each tangent equal to QO. 

(i.) The equation of the chord of contact, for the point 

whose vector is 

p=pa + q^y 

Bt ^ 
is thus p = a^i+-y--fy(a-fj»^).. 

D 2 

20 QUATERNIONS. . [chap. I. 

Suppose this to pass always through the point whose vector is 

p = aa-\-b^. 
Then we must have 

^1 4-y = a, 

or ^1 =jo f >yj»* — 2j9fl + 2i. 

Comparing this with the expression in (^), we have 

that is^ the point from which the tangents are drawn has the 

= —bfi^p{a-\-a^), a straight line (§ 28 (1)). 

The mere form of this expression contains the proof of the usual 
properties of the pole and polar in the parabola; but, for the 
sake of the beginner, we adopt a simpler, though equally general, 

Suppose a = 0, This merely restricts the pole to the par- 
ticular diameter to which we have referred the parabola. Then 
the pole is Q, where p =r bB- 

and the polar is the line TU, for which 
p =-'bP'\-pa. 
Hence tke polar of any point is parallel to the tangent at the 
extremity of the diameter on which the point lies, and its inter^ 
section with that diameter is as far beyond the vertex as the^poU is 
within, and vice versa. 

(J,) As another example let us prove the following theorem. 
I/^ a triangle be inscribed in a parabola, the three points 
in which the sides are met by tangents at the angles He 
in a straight line. 

Since is any point of the curve, we may take it as jone 


corner of the triangle. Let t and t^ determine the others. 
Then, if w^, Wa, m^ represent the vectors of the points of inter- 
section of the tangents with the sides, we easily find 



. 2t-t 




These vahies 









U ' t ' tt. 

CTa = 0. 

Also -— — — ^ — — = identically. 

Hence, by § 30, the proposition is proved. 

(^.) Other interesting examples of this method of treating 

curves will, of course, suggest themselves to the 

student. Thus 

p = a cos ^ -f j3 sin ^ 

or p = ax-\-fi\/\^x'^ 

represents an ellipse, of which the given vectors a and /3 are 

semi-conjugate diameters. 


pc=at-^y or p = atana? + /3coti?? 

evidently represents a hyperbola referred to its asymptotes. 

But, so far as we have yet gone, as we are not prepared 
to determine the lengths or inclinations of vectors, we can only 
investigate a very small class of the properties of curves, re- 
presented by such equations as those above written. 

(L) We may now, in extension of the statement in § 29, 
make the obvious remark that 
p = 2pa 


is the equation of a curve in space^ if the numbeife j»i, jOj, &e. 
are functions of one indeterminate. In such a ease the equation, 
is sometimes written ^ _ a^u)^ 

But, i(jpi,pty &e. be functions of two indeterminates, the locus 
of the extremity of p is a surface ; whose equation is sometimes 
written ^^^(j^^^y 

(m.) Thus the equation 

p = acos^ + /3sin^-|-y^ 

belongs to a helix. 

Again, p =J»a + y^ + ry 

with a condition of the form 

ajo^-f ^j»-f (?r^ = 1 

belongs to a central surface of the second order, of which a, j8, y 
are the directions of conjugate diameters. If «, i, c be all 
positive, the surface is an ellipsoid. 

32. In Example {/) above we performed an operation equi- 
valent to the diflPerentiation of a vector with reference to a single 
numerical variable oT which it was given as an explicit fimction. 
As this process is of very great use, especially in quaternion 
investigations connected with the motion of a particle or point ; 
and as it will aflPord us an opportunity of making a preliminary 
step towards overcoming the novel difficulties which arise in 
quaternion differentiation; we will devote a few sections to a 
more careful exposition of it. 

33. It is a striking circumstance, when we consider the way 
in which Newton^s original methods in the Differential Calculus 
have been decried, to find that Hamilton was obliged to employ 
them, and not the more modem forms, in order to overcome the 
characteristic difficulties of quaternion diflferentiation. Such 
a thing as a differential coefficient has absolutely no meaning in 


quaternions, except in those special cases in which we are dealing 
with degraded quaternions, such as numbers, Cartesian coor- 
dinates, &c. But a quaternion expression has always a differ^ 
entialy which is, simply, what Newton called 2i fluxion. 

As with the Laws of Motion, the basis of Dynamics, so with 
the foundations of the Differential Calculus ; we are gradually 
coming to the conclusion that Newton's system is the best 
after all. 

34. Suppose p to be the vector of a curve in space. Then, 
generally, p . may be expressed as the sum of a number of 
terms, each of which is a multiple of a given vector by a function 
of some one indeterminate ; or, as in § 3 1 (/), if P be a point on 
the curve, 6P = pz=<p{f). 

And, similarly, if Q be any other point on the curve, 

OQ = p, = <p{t,)^<l>{t^btl 
where bt is any number whatever. 

The vector-chord PQ is therefore, rigorously, 
hp = pi — p = <^(t-^ht) — ^t, _ 

35. It is obvious that, in the present case, because the vectors 
involved in (j> are constant, and their numerical multipliers alone 
vary, the expression <f){t-{-hf) is, by Taylor's Theorem, equivalent to 

^(^ + -__8^+_^__ + 

Hence; ,^^mu+'-l^ ^ ^ ^o. 
^ dt dp 1.2 

And we are thus entitled to write, when ht has been made in- 
definitely small. 

In such a case as this, then, we are permitted to diflferentiate. 



[chap. I. 

or to fonn the diflferential coefficient of, a vector according to 
the ordinary rules of the Differential Calculus. But great ad- 
ditional insight into the process is gained by applying Newton^s 

36. Let OP be 

and OQi 

where dt is any number what- 

The number t may here be 
taken as representing timey i. e. 
we may suppose a point to move 
along the curve in such a way 
that the value of t for the vector of point P of the curve de- 
notes the interval which has elapsed (since a fixed epoch) when 
the moving point has reached the extremity of that vector. If, 
then, dt represent any interval, finite or not, we see that 


will be the vector of the point after the additional interval dt. 

But this, in general, gives us little or no information as to 
the velocity of the point at J^. We shall get a better approxi- 
mation by halving the interval dty and finding Q^, where 
OQa = <^(^4-4rf^, as the position of the moving point at that 
time. Here the vector virtually described in i^/J is PQa. To 
find, on this supposition, the vector described in dt, we must 
double this, and we find, as a second approximation to the vector 
which the moving point would have described in time dt, if it 
had moved for that period in the direction and with the velocity 
it had at P, 

P^, = 2PQ, = 2(OQ,-aP) 

= 2{<^(^ + i^0-<^^}- 


The next approximation gives 

^3 = 3PQ, = 3(OQ3-QP) 

And so on^ each step evidently leading ns nearer the sought truth. 
Hence, to find the vector which would have been described in 
time dt had the circumstances of the motion at P remained 
undisturbed, we must find the value of 

We have seen that in this particular case we may use Taylor's 
Theorem. We have, therefore, 

dp = ^x=«^{*'W^rf^+*''(0^ ^ +&e.} 
= 4>\()dt. 
And, if we choose, we may now write 

37. But it is to be most particularly remarked that in the 
whole of this investigation no regard whatever has been paid to 
the magnitude of dt. The question which we have now answered 
may be put in the form — A point describes a given curve in a 
given manner. At any paint of its path its motion suddenly ceases 
to be accelerated. What space will it describe in a definite interval ? 
As Hamilton well observes, this is, for a planet or comet, the 
case of a '' celestial Attwood's machine.^' 

38. If we suppose the variable, in terms of which p is ex- 
pressed, to be the arc, «, of the curve measured from some fixed 
point, we find as before 


= ^'(s)ds. 



From the veiy nature of the question it is obvious that the 
length of dp must in this case be ds. This remark is of im- 
portance^ as we shall see later ; and it may therefore be useful 
to obtain afresh the above result without any reference to time 
or velocity. 

89. Following strictly the process of Newton's Vllth Lemma, 
let us describe on Pq^ an arc similar to PQ^, and so on. Then 
obviously, as the subdivision of ds is carried farther, the new arc 
(whose length is always ds) more and more nearly coincides with 
the line which expresses the corresponding approximation to dp, 

40. As a final example let us take the hyperbola 

p = at+^. 
Here . fi. 

This shews that the tangent is parallel to the vector 

In words, if ike vector {from the centre) of u point in a hyperbola 
be one diagonal of a parallelogram^ two of whose sides coincide 
with the asymptotes^ the other diagonal is parallel to the tangent 
at the point. 

41. Let us reverse this question, and seek the envelop of a 
Une which cuts off from two fixed axes a triangle <f constant area. 

If the axes be in the directions of a and /8, the intercepts 

may evidently be written at and y . Hence the equation of the 
line is (§ 30) 

p = at-^xy^ — aty 


The condition of envelopment is, obviously, 

dp = 0. 


This gives 

= {a-a?(-^ + a)}dt-^(^-at)dx. 

[We are not here to equate to zero the eoeffieienta of dt and dx ; 
for we must remember that this equation is of the form 

=i?a + g'/3, 

wliere j9 and q are numbers ; and that^ so long as a and j3 are 
actual and non-parallel vectors^ the existence of suoh an equation 
requires i? = 0, ? = 0.] 

Hence {l'-x)dt'-tdx = 0, 

and —^^^j. _ = 0. 

t* ^ t 

From these, at once, x-^^, since dx and dt are indeterminate. 
Thus the equation of the envelop is 

the hyperbola as before ; o, jS being portions of its asymptotes. 

42. It may assist the student to a thorough comprehension 
of the above process, if we put it in a slightly different form. 
Thus the equation of the enveloping line may be written 

p = aj{(l-a?)+i3|, 

which gives dp = = ad{t{\ —a?)) +/3rf (-) . 

Hence, as a is not parallel to j3, we must have 

and these are, when expanded, the equations we obtained in the 
preceding section. 

E % 

28 QUATBRNIOKS. [ohap. L 

43. For farther illastration we give a solution not directly 
employing the differential calcnius. The equations of any two 
of the enveloping lines are 


t and t^ being given^ while s and y are indeterminate. 
At the point of intersection of these lines we have (§ 26), 

^(l-ar) = ^,(l-y),- 

^ — JL 
t t^ * 

These give, by eliminating y, 

or X "^ • 

Hence the vector of the point of intersection is 

and thus, for the ultimate intersections, where j^ —- = 1, 


p = \{at-\-^ as before. 

CoE. (1). If «, = 1, 

/, = — , 

or the intersection lies in the diagonal of the parallelogram on 

Cor. (2). If t^ = mtj where m is constant, 


But we have also x = — • 

Hence the locus of anoint which divides in a given ratio a line 
cutting off a given area from two fixed axes, is a hyperbola of which 
these axes are the asymptotes. 

Cor. (3). If we take 

tt^\t'\-t^ = constant 
the locus is a parabola ; and so on. 

44. The reader who is fond of Anharmonic Ratios and Trans- 
versals will find in the early chapters of Hamilton's Elements 
of (Quaternions an admirable application of the composition of 
vectors to these subjects. The Theory of Geometrical Nets, in 
a plane, and in space, is there very folly developed; and the 
method is shewn to include, as particular cases, the processes of 
Orassmann's Ausdehnungslehre and Mobius' Baryeentrische CalcuL 
Some very curious investigations connected with curves and 
sar&ces of the second and third orders are also there founded 
apon the composition of vectors. 


1. The lines which join, towards the same parts, the ex- 
Iremities of two equal and parallel lines are themselves equal 
and parallel. {Huclid, I. xxxiii.) 

2. Knd the vector of the middle point of the line which joins 
the middle points of the diagonals of any quadrilateral, plane 
or gauche, the vectors of the comers being given ; and so prove 
that this point is the mean point of the quadrilateral. 


If two opposite sides be divided proportionally, and two new 
quadrilaterals be formed by joining the points of division, the 
mean points of the three quadrilaterals lie in a straight line. 

Shew that the mean point may also be found by bisecting 
the line joining the middle points of a pair of opposite sides. 

3. Verify that the property of the coefficients of three vectors 
whose extremities are in a line* (§ 30) is not interfered with by 
altering the origin. 

4. If two triangles ABC, abc, be so situated in space that 
Jjdf Bb, Cc meet in a point, the intersections of AB^ ab, of BC, 
be, and of CA, ca, lie in a straight line. 

5. Prove the converse of 4, i. e. if lines be drawn, one in 
each of two planes, from any three points in the straight line 
in which these planes meet, the two triangles thus formed are 
sections of a common pyramid. 

6. If five quadrilaterals be formed by omitting in succession 
each of the sides of any pentagon, the lines bisecting the 
diagonals of these quadrilaterals meet in a point. (H. Fox 

7. Assuming, as in § 7, that the operator 

cos^+ V—l sinO 

turns any radius of a given circle through an angle in the 
positive direction of rotation, without altering its length, deduce 
the ordinary formulae for cos {A-{-B), cos (A^B), sin {A-\-B), and 
sin {A—B)y in terms of sines and cosines of A and B. 

8. If two tangents be drawn to a hyperbola, the line joinmg 
the centre with their point of intersection bisects the lines join- 
ing the points where the tangents meet the asymptotes : and 
the tangent at the point where it meets the curves bisects the 
intercepts of the asymptotes. 

9. Any two tangents, limited by the asymptotes, divide each 
other proportionally. 


10. If a chord of a hyperbola be one diagonal of a paralM- 
ogram whose sides are parallel to the asymptotes, the other 
diagonal passes through the centre. 

11. Show that 

is the equation of a cone of the second degree, and that its 
section by the plane 

is an ellipse which touches, at their middle points, the sides of 
the triangle of whose corners a, j3, y are the vectors. (Hamilton, 
Elements y p. 96.) 

12. The lines which divide, proportionally, the pairs of opposite 
sides of a gauche quadrilateral, are the generating lines of a 
hyperbolic paraboloid. (Ibid, p. 97.) 

13. Show that 

where x-^y-\-z = 0, 

represents a cone of the third order, and that its section by 
the plane 

is a cubjc curve, of which the lines 

are the asymptotes and the three (real) tangents of inflexion. 
Also that the mean point of the triangle formed by these lines 
is a conjugate point of the curve. Hence that the vector 
^-Hi3 + y is a conjugate ray of the cone. {Ibid. p. 96.) 



46. Xl/E now come to the consideration of points in 
▼^ which the Calculus of Quaternions differs entirely 
from any previous mathematical method; and here we shall get 
an idea of what a Quaternion is^ and whence it derives its name. 
These points are Aindamentally involved in the novel use of the 
symbols of multiplication and division. And the simplest in- 
troduction to the subject seems to be the consideration of the 
quotient^ or ratio^ of two vectors. 

46. If the given vectors be parallel to each other^ we have 
already seen (§ 22) that either may be expressed as a numerical 
multiple of the other; the multiplier being simply the ratio 
of their leng^hs^ taken positively if they are similarly directed^ 
negatively if they run opposite ways. 

47. If they be not parallel^ let OA and OB be drawn parallel 
and equal to them from any point 0; and the question is re- 
duced to finding the value of the ratio of two vectors drawn 
from the same point. Let us try to find tipon how many^ distinct 
numbers this ratio depends* 

We may suppose OA to be changed into OB by the following 

1st. Increase or diminish the length of OA till it becomes 
equal to that of OB, For this only one number is 
required, viz. the ratio of the lengths of the two 
vectors. As Hamilton remarks, this is a positive, or 
rather a signless^ number. 

2nd. Turn OA about until its direction coincides with 
that of OB, and (remembering the effect of the first 


operation) we see that the two vectors now coincide 

or become identical. To specify this operation three 

more numbers are required^ viz. two angles (such as 

node and inclination in the case of a planet's orbit) 

to fix the plane in which the rotation takes place^ 

and one angle for the amount of this rotation. 

Thus it appears that the ratio of two vectors, or the multiplier 

required to change one vector into another, in general depends 

w^oufoTJi/r distinct numbers, whence the name quaternion. 

48. It is obvious that the operations just described may be 
performed, with the same result, in the opposite order, being 
perfectly independent of each other. Thus it appears that a 
quaternion, considered as the factor or agent which changes one 
definite vector into another, may itself be decomposed into two 
factors of which the order is immaterial. 

The stretching factor, or that which performs the first opera- 
tion in § 47, is called the Tensor, and is denoted by prefixing 
T to the quaternion considered. 

The turning &ctor, or that corresponding to the second opera^ 
tion in § 47, is called the Vbbsor, and is denoted by the letter U 
prefixed to the quaternion. 

49. Thus, if 0A = a, OB = /8, and if j be the quaternion 
which changes o to )8, we have 

which we may write in the form 

^ = ?> or pa" = q, 

if we agree to defifie that 

-.a = i8a-».a = /8. 

Here it is to be particularly noticed that we write q before a to 

signify that a is multiplied by q, not q multiplied by o. 

[This remark is of extreme importance in quaternions, for. 


as we shall soon see^ the Commutative Law does not generally 
apply to the factors of a product.] 

We have also, by § 47, 

where, as before, Tq depends merely on the relative lengths of 
a and j3, and Uq depends solely on their directions. 

Thus, if tti and p^ be vectors of unit length parallel to a and 
^ respectively, R R 

a. at a 

50. We must now carefully notice that the quaternion which 
is the quotient when fi is divided by a in no way depends upon 
the absolute lengths, or directions, of these vectors. Its value 
will remain unchanged if we substitute for them any other pair 
of vectors which 

(1) have their lengths in the same ratio, 

(2) have their common plane the same or parallel, 
and (3) make the same angle with each other. 

Thus in the annexed figure 

if, and only if, 

,^. 0,B, OB 

(2) plane A OB parallel to plane 

(3) lAOB- LA^O^By, 

[Equality of angles is understood to in- 
clude similarity in direction. Thus the ro- 
tation about an upward axis is negative (or right-handed) from 
OA to OBy and also from O^A^ to OyB^:\ 


5L The Reciprocal of a quaternion q is defined by the equation, 

Hence if 

- = ^, or 

we must have - = - = g-^. 

For this gives ^ • ^ = S^^ • ?^> 


and each side is evidently equal to a. 

Or, we may reason thus, q changes OA to OB, qr^ must there- 
fore change 0£ to OA, and is therefore expressed by - (§ 49). 

The tensor of the reciprocal of a quaternion is therefore the 
reciprocal of the tensor ; and the versor differs merely by the 
reversal of its representative angle. 

52. The Conjugate of a quaternion q, written Kq, has the same 
tensor, plane, and angle, only the angle is taken the reverse 
way. Thus, if 

OA'^OA, and lAOB^ LAOB, 

\B OB OB . \ r JT 

-T= = a, ^r^ir- = coniuffate oi q^Kq, 

OA ^ OA' ^ ^ 

By last section we see that 
Kq.= {Tqyq-\ 
Hence qKq = Kq,q = {Tq)K 
This proposition is obvious, if we recollect 
that the tensors of q and Kq are equal, and 
that the versors are such that either reverses the effect of the 
other. The joint effect of these factors is therefore merely to 
multiply twice over by the common tensor. 

63. It is evident from the results of § 50 that, if a and /3 

F % 



[chap. II. 

be of equal length, their quaternion quotient becomes a versor 
(the tensor being unity) and may be represented indifferently 
by any one of an infinite number of arcs of given length lying 
on the circumference of a circle, of which the two vectors are 
radii. This is of considerable importance in the proofs which 

Thus the versor -r= may be repre- 
sented by the arc AB, which may be 
written AB, 

And, similarly, the versor ^ is 

represented by A^B^ which is equal to 

(and measured in the same direction as) AB if 

LAfiB^^ LAOB, 
i. e. if the versors are equal. 

54. By the aid of this process, when a versor is represented 
by an arc of a great circle on the unit-sphere, we can easily 
prove that quaternion multiplication is not generally commutative. 

Thus let q be the versor AB or 

OB "^ '-^ 

-^=T, Make BC-=- AB, then q may 



also be represented by -r^r 

In the same way any other versor 

r may be represented by DB or BE 

^^ OB 6E 
and by -= or -n=r . 
^ OB OB 

The line 0^ in the figure is definite, and is given by the 

intersection of the planes of the two versors ; being the centre 

of the unit-sphere. , 

Now rOD = OB, and qOB = OC, 

Hence qrOB^OC, 


OC ^-N 

or qr = -=■ , and may therefore be represented by the are JDC 

of a great circle. 

But rq is easily seen to be represented by the arc AE, 
For qOA= OB, and r05 = OE, 


whence rqOA = OE, and rq = -==■• 


Thus the versors rq and qr, though represented by arcs of equal 

length, are not generally in the same plane and are therefore 

unequal : unless the planes of q and r coincide. 

Calling OA a, we see that we have assimied, or defined, in 

the above proof, that q.ra = qr.a and r,qa = rq,a when qa, ra, 

qra, and rqa are all vectors. 

55. Obviously C£ is Kq, BB is Kr, and CB is K{qr). But 
CB^BB.CBy which gives us the very important theorem 

K{qr) = Kr.Kq. 

56. The propositions just proved are, of course, true of 
quaternions as weU as of versors ,• for the former involve only 
an additional numerical factor which has reference to the length 
merely, and not the direction, of a vector (§ 48). 

57. Seeing thus that the commutative law does not in general 
hold in the multiplication of quaternions, let us enquire whether 
the Associative Law holds. That is, if jo, j, r be three quaternions, 
have we p .qr •=. ;pq,rl 

This is, of course, obviously true if jo, q, r be numerical quantities, 
or even any of the imaginaries of algebra. But it cannot be 
considered as a truism for symbols which do not in general give 

58. In the first place we remark that jo, q, and r may be 
considered as versors only, and therefore represented by arcs of 

38 QUATERNIONS. [chap. II. 

great circles, for their tensors may obviously /(§ 48) be divided 
out from both sides, being commutative with the versors. 

Let AB =jo, m)=^c2 = q,2JidI^=r. 

Join BC and produce the great circle till it meets JEF in ^, 
BXidmake KH=F£=r,mdHG = CB=pq (§54). 

Join GK. Then 


Join FD and pro- 
duce it to meet AS 
in M, Make 

and MN=AB, 
^^ and join NL, Then 

Hence to shew that p.qr =pq,r 
all that is requisite is to prove that LN, and KG, described as 
above, are equal arcs of the same great circle, since, by the figure, 
the}'^ are evidently measured in the same direction. This is 
perhaps most easily effected by the help of the fundamental 
properties of the curves known as Spherical Conies, As they are 
not usually familiar to students, we make a slight digression for 
the purpose of proving these fundamental properties; after 
Chasles, by whom and Magnus they were discovered. An 
independent proof of the associative principle will presently be 
indicated, and in Chapter VII we shall employ quaternions to 
give an independent proof of the theorems now to be established. 

59.* Dbf. a spherical conic is the curve of intersection of a 
cone of the second degree with a sphere, the vertex of the cone being 
the centre of the sphere. 

If a cone have one series of circular sections, it has another 
series, and any two circles belonging to different series lie on a 
sphere. This is easily proved as follows. 

Describe a sphere. A, cutting the cone in one circular section. 



C, and in any other point whatever, and let the side OpP of the 
cone meet A in p, P ; P being a point in C. Then PO'Op is 
constant, and, therefore, since P lies in a plane, p lies on a 
sphere, «, passing through 0. Hence the locus, c, of js is a 
circle, being the intersection of the two spheres A and a. 

Let OqQ be ^ny other side of the cone, q and Q being points 
in c, C, respectively. Then the quadrilateral q QPp is inscribed 
in a circle (that in which its plane cuts the sphere A) and the 
exterior angle at p is equal to the interior angle at Q. If OX, 
OM be the lines in which the plane POQ cuts the cyclic plants 
(planes through parallel to the two series of circular sections) 
they are obviously parallel to pq, QP^ respectively ; and therefore 

lLOp = lOpq=lOQPz= IMOQ. 

Let any third 

side, OrE, of 
the cone be 
drawn, and let 
the plane OPS 
cut the cyclic 
planes in 01, 
Om respective- 
ly. Then, evi- 
dently, 110L= A qpr, L MOm = Z QPR, 
and these angles are independent of the position of the points 
p and Py if Q and R be lixed points. 

In a section of the above diagram by a sphere whose centre 

is 0, IL, Mm are the 
great circles which re- 
present the cyclic planes, 
PQR is the spherical 
conic which represents 
the cone. The point P 
represents the line OpP, 
and so with the others. 


The propositions above may now be stated thus 
ArcPi = arcJfQ; 

and, if Q and E be fixed, Mm and IL are constant arcs whatever 
be the position of P. 

60. The application to § 58 is now obvious. In the fig^e 
of that article we have 


Hence i, C, G, D are points of a spherical conic whose cycKc 
planes are those of AB, FE, Hence also KG passes through L, 
and with LM intercepts on AB an arc equal to AB. That is, 
it passes through N, or KG and LN are arcs of the same great 
circle : and they are equal, for G and L are points in the spherical 

Also, the associative principle holds for any number of quater- 
nion &ctors. For, obviously, 

qr,8t=qrs,t, &c., &c.^ 

since we may consider ^r as a single quaternion, and the above 
proof applies directly. 

6L That quaternion addition, and therefore also subtraction, 
is commutative, it is easy to shew- 

For if the planes of two qua- 
ternions, q and r, intersect in the 
line OA^ we may take any vector 
OA in that line, and at once find 
two others, O^and 067, such that 

and 00 = rOA. 

And {q+r)6A = 6B+0C= 0C+ OB = (r+j)Q2, 

since vector addition is commutative (§ 27). 

Here it is obvious that {q-^-r) OAy being the diagonal of the 
parallelogram on OB, OC, divides the angle between OB and OC 



in a ratio depending solely on the ratio of the lengths of these 
lines^ i. e. on the ratio of the tensors of q and r. This will be 
usefiil to us in the proof of the distributive law, to which we 

62. Quaternion multiplication, and therefore division, is dis- 
tributive. One simple proof of this depends on the possibility, 
shortly to be proved, of representing any quaternion as a linear 
function of three given rectangular unit-vectors. And when 

the proposition is thus 
established, the asso- 
ciative principle may 
readily be deduced from 

But we may employ 
for its proof the proper- 
ties of Spherical Conies 
already employed in de- 
monstrating the truth of 
the associative principle. 
For continuity we give an outline of the proof by this process. 

Let £Ay CA represent the versors of q and r, and be the great 
circle whose plane is that of J9. 

Then, if we take as operand the vector OJ, it is obvious that 
U[qj^f^ will be represented by some such arc as DA where 
B,D, C sj^e in one great circle; for {q-^-rjOA is in the same 
plane as qOA and rOA, and the relative magnitudes of the arcs 
BS and DC depend solely on the tensors of q and r. Produce 
BA, DA, CA to meet be in i, d, c respectively, and make 

M = £A, Fd=:DA, Gh = CA. 

Also make J^=58=cy =j». Then B, P, G, A lie on a spherical 
conic of which BC and be are the cyclic arcs. And, because 
ij3=35=cy, j3^, bF, yG, when produced, meet in a point H 


which is also on the spherical conic (§ 59*). Let these arcs meet 
BC mJyL,K respectively. Then we have 

Also U^BI), 

and KL=^CI>. 

Andj on comparing the portions of the figure bounded respect- 
ively by HKJ and by ACB we see that (when considered with 
reference to their effects as factors multiplying OH and OA 

P ^(?+ bears the same relation to pUq and p Ur 

that U{q+r) bears to Uq and Ur. 

But T{q + r) U{q + r) = q + r = TqUq^-Trllr. 

Hence T{q -f r),p U{q + r) = Tq.p Uq-^-Tr.p Ur ; 

or, since the tensors are mere numbers and commutative with 

all other factors, 

p{q-\'r) —pq+pr. 

In a similar manner it may be proved that 

{q + r)p ^zqp-^-rp. 

And then it follows at once that 

{p-\-q){r-\-8) ^pr-^-ps+qr+qs. 

63. By similar processes to those of § 53 we see that 
versors, and therefore also quaternions, are subject to the index- 
law ^^ ^„ _ ^^+^^ 

at least so long as m and n are positive integers. 

The extension of this property to negative and fractional 


exponents must be deferred until we have defined a negative 
^r fractional power of a quaternion. 

64. We now proceed to the special case of quadrantal versors, 
from whose properties it is easy to deduce all the foregoing 
results of this chapter. These properties were indeed those 
whose discovery by Hamilton in 1843 led almost intuitively 
to the establishment of the Quaternion Calculus. We shall 
eontemb ourselves at present with an assumption, which will 
be shewn to lead to consistent results ; but at the end of the 
chapter we shall shew that no other assumption is possible, 
following for this purpose a very curious quasi-metaphysical 
speculation of Hamilton's. 

65. Suppose we have a system of three mutually perpen- 
dicular unit-vectors, drawn from one point, which we may call 
for shortness 7, /, K, Suppose also that these are so situated 
that a positive (i. e. left-handed) rotation through a right angle 
about I as an axis brings / to coincide with K, Then it is 
obvious that positive quadrantal rotation about / will make K 
coincide with I; and, about K, will make I coincide with /. 

For definiteness we may suppose / to be drawn eastwards^ 
J northwards y and K upwards. Then it is obvious that a positive 
(left-handed) rotation about the eastward line (7) brings the 
northward line (/) into a vertically upward position {K) ; and 
so of the others. 

66. Now the operator which turns / into £' is a quadrantal 
versor (§ 53) ; and, as its axis is the vector 7, we may call it i. 

Thus ^ = h or K=iJ. (1) 


Similarly we may put 

-j^=y, or 7=:yZ, (2) 

and '-j—hy or J—kL (3) 

G 1 


[It may here be noticed^ merely to shew the symmetry of 
the system we are explaining^ that if the three mutnally per- 
pendicular vectors /, /, Jf be made to revolve about a line 
equally inclined to all, so that / is brought to coincide with /, 
/ will then coincide with K, and K with 7: and the above 
equations will still hold good, only (1) will become (2), (2) will 
become (3), and (3) will become (1).] 

67. By the results of § 50 we see that 

-/_ K 

K ^ J' 
i. e. a southward unit- vector bears the same ratio to an upward 
unit-vector that the latter does to a northward one ; and there- 
fore we have 

— / .• 

-— = i, or — /=»£ (4) 


Similarly ~^^' ^^ —K=JI; (5) 

and —=-=:*, or ^I=lkJ. (6) 

68. By (4) and (1) we have 

-./=ije'=:i(ic/) = i»/. 

Hence i» = — 1 (7) 

And, in the same way, (5) and (2) give 

>'=-!. (8) 

and (6) and (3) ^^^^ ^^^ 

Thus, as the directions of 7, e/", K are perfectly arbitrary, we 
see that tAe square of every quadrantal veraor is negative unity. 

[Though the following proof is in principle exactly the same 
as the foregoing, it may perhaps be of use to the student, in 
shewing him precisely the nature as well as the simplicity of 
the step we have taken. 

A' -OC 


Let ABjt be a semicircle^ whose 
centre is 0, and let OB be perpen- 
dicular to AOA\ 

Inen -=-, = q suppose^ is a qua- 

drantal versor, and is evidently equal 


to ^; §§50,53. 

„ ^ 62 OB 62 _ 

Hence q^ = -=•• ^=^ = •^=^ = — 1. 1 

^ 05 J OA -" 

69. Having thus found that the squares of iyj, k are each 
equal to negative unity ; it only remains that we find the values 
of their products two and two. For, as we shall see, the result 
is such as to shew that the value of any other combination whfit- 
ever of i,j, k (as factors of a product) may be deduced from the 
values of these squares and products. 

Now it is obvious that 

(i. e. the versor which tiuns a westward unit- vector into an 
upward one will turn the upward into an eastward unit) ; 

or JE'=y(-/)=-y7. (10) 

[The negative sign, being a mere nimierical factor, is evidently 
commutative with^; indeed we may, if necessary, easily assure 
ourselves of the fact that to turn the negative (or reverse) of 
a vector through a right (or indeed any) angle, is the same 
thing as to turn the vector through that angle and then re- 
verse it.] 

Now let us operate on the two equal vectors in (10) by the 
same versor, i, and we have 

iK=.i {—jl) = —iJL 
But by (4) and (3) 



[chap. II. 

Comparing these equations we have 


or, hy § 54 (end), ij = i, " 

and symmetry g^ves jh ^ i, - (11) 

M = J. 

The meaning of these important equations is very simple; 
and is, in fact, obvious from our construction in § 54 for the 
multiplication of versors; as we see by the annexed figure, 
where we must remember that i, J, k are quadrantal versors 
whose planes are at right angles, so that the figure represents a 

hemisphere divided into quadrantal 

Thus, to show that ij = i, we 
have, being the centre of the 
sphere, iV", ^, Sy F'the north, east, 
south, and west, and Z the zenith 
(as in § 65) ; 



iJOW^idZ^z OS^IOW. 

70. But, by the same figure, 
whence ji ON = jOZ ^OE z:z-^6W ^-kON. 

7L From this it appears that 
and similarly hj — — iy ► (12) 

ih =— y,^ 

and thus, by comparing (11), 

jk^-kj= iy I ((11), (12)). 
ki ^—ik — J, 


These equations, along with 

i2=y2 = ^2=_l ((7), (8), (9)), 

eontain essentially the whole of Quaternions. *But it is easy 
to see that, for the first group, we may substitute the single 

equation ijk=.-\, (13) 

since from it, by the help of the values of the squares of i,y, k, 
all the other expressions may be deduced. We may consider 
it proved in this way, or deduce it afresh from the figure above, 

k6N= OW, 

jhON^ jOW^ OZy 

ijkm = ij6W^iOZ=:08^-^6N. 

72. One most important step remains to be made, to wit the 
assumption referred to in § 64. We have treated i,y, k simply 
as quadrantal versors; and I, J, K 2iB unit- vectors at right 
angles to each other, and coinciding with the axes of rotation 
of these versors. But if we collate and compare the equations 
just proved we have 

■ = h (11) 

!»./■= K, (1) 

\ji=-k, (12) 

ijJ=-K, (10) 

with the other similar groups symmetrically derived from 
them. Now the meanings we have assigned to i,/, k are 
quite independent of, and not inconsistent with, those assigned 
to 7, J, K. And it is superfluous to use two sets of characters 
when one will suffice. Hence it appears that i,j, k may be 
substituted for lyJ^K; in other words, a unit-vector when em- 
ployed as a factor may be considered as a quadrantal versor whose 
plane is perpetidicular to the vector. This is one of the main 
elements of the singular simplicity of the quaternion calculus. 


78. Thus tie product, and therefore the quotient, of two per- 
pendicular vectors is a third vector perpendicular to both. 

Hence the reciprocal (§ 51) of a vector is a vector wliich has. 
the opposite direction to that of the vector, and its length is the 
reciprocal of the length of the vector. 

The conjugate (§ 52) of a vector is simply the vector reversed. 

Hence, by § 52, if a be a vector 

{Taf = aKa = a(-a) = -a'. 

74. We may now see that every versor may be represented by 
a power of a unit-vector. 

For, if a be any vector perpendicular to i (which is any definite 

ia , = ^, is a vector equal in length to a, but perpendicular 
to both i and a ; 

i^ a = — a, 

^'a =— ia =— /3, 

i*a = — i)3 = — i»a = a. 

Thus, by successive applications of i, a is turned round ^ as an 
axis through successive right angles. Hence it is natural to 
defme i^ as a versor which turns any vector perpendicular to i 
through m right angles in the positive direction of rotation about 
i as an axis. Here m may have any real value whatever, for 
it is easily seen that analogy leads us to interpret a negative 
value olmas corresponding to rotation in the negative direction. 

75. From this again it follows that any quaternion may be 
exvressed as a power of a vector. For the tensor and versor 
elements of the vector may be so chosen that, when raised to 
the same power, the one may be the tensor and the other the 
versor of the given quaternion. The vector must be, of course, 
perpendicular to the plane of the quaternion. 

76. And we now see, as an immediate result of the last two 



sections^ that the iudex-law holds with regard to powers of a 
quaternion (§ 63). 

77. So far as we have yet considered it, a quaternion has 
been regarded as the product of a tensor and a versor : we are 
now to consider it as a %um. The easiest method of so analysing 
it seems to be the following. 

Let -=■ represent any quaternion. 

Draw BC perpendicular to OA^ pro- 
duced if necessary. 

Then, § 19, Qfi = OC+CB. 
But, § 22, 0C=x02 

where a: is a number, whose sign is 
the same as that of the cosine of 
Also, § 73, since CB is perpendicular to OAy 

where y is a vector perpendicular to OA and CB, i. e. to the 
plane of the quaternion. 

OB xOA + yOA 




= a?+y. 

Thus a quaternion, in general, may be decomposed into the 
sum of two parts, one numerical, the other a vector. Hamilton 
calls them the scalae, and the vector, and denotes them respect-* 
ively by the letters S and F prefixed to the expression for the 

78. Hence q=: SqA- Vq, and if in the above example 


= ?. 

then OB^OC-^ CB^ Sq.OA+Vq. OA. 


[The points are inserted to shew that S and F apply only to q, 
and not to qOAJ] 

The equation above gives 


CB = Vq.OA. 

79. If, in the figure of last section, we produce BC to 2)^ so 
as to double its lengthy and join OL, we have by § 52^ 


.-. 00= 0C+ CD = SKq,02 + FKq.OA, 
Hence 0C=8Kq.02, 

and CB^ FKq.OA. 

Comparing this value of OC with that in last section, we find 

SKq^Sq, (1) 

or the scalar of the conjugate of a quaternion is eqnal to the scalar 
of the quaternion, 

Agaiuj CB= — CB by the figure, and the substitution of their 
values gives FKq = --Fq, (2) 

or the vector of the conjugate of a quaternion is the vector of the 
quaternion reversed. 

[We may remark that the results of this section are simple 
consequences of the fact that the symbols S, F, K are com- 
mutative*. Thus 

SKq = KSq = 8q, 

since the conjugate of a number is the number itself; and 
FKq = KFq = ^Fq {^ Vs). 

* It iB curious to compare the properties of these quaternion symbols with 
those of the Elective Symbols of Logic, as given in Boole's wonderful treatise 
on the Lcbws of Tk(mght ; and to think that the same grand science of mathematical 
analysis, by processes remarkably similar to each other, reveals to us truths in 
the science of position fiur beyond the powers of the geometer, and truths of 
deductive reasoning to which unaided thought could never have led the lo- 


Again, it is obvious that 

iSq = 82q, iFq^ Flq, 
and thence ^Kq = K:Lq,'] 

80. Since any vector whatever may be represented by 

where x, y, z are numbers (or Scalars), and i, jy k may be any 
three non-coplanar vectors, §§23, 25 — though they are usually 
understood as representing a rectangular system of unit- vectors — 
and since any scalar may be denoted by «? ; we may write, for 
any quaternion q, the expression 

q = w-\-xi-\-yj'\-zh (§ 78). 

Here we have the essential dependence on four distinct 
numbers, from which the quaternion derives its name, exhibited 
in the most simple form. 

And now we see at once that an equation such as 

where ^=w'-^'X^i-\-/j-\-zk, 

involves, of course, the four equations 

w'= w, a?'= Xy y = y, / = z. 

8L We proceed to indicate another mode of proof of the 

distributive law of multiplication. 

We have already defined, or assumed (§ 61), that 

P y /3 + y 

— I — = i 

a a a 

or fia-'A-ya-'^ = {p-^y)a-'', 
and have thus been able to understand what is meant by adding 
two quaternions. 
But, writing a for a-^ we see that this involves the equality 
(i3-hy)o = ^a 4- ya; 
from which, by taking the conjugates of both sides, we derive 
a'(i8'+/) = a'^+aV (§55). 

H % 


And a combination of these results (putting /3-f-y for a in the 
latter^ for instance) g^ves 

(/3-fy)(^+/) = 0+y)/3' + (i8+y)/ 

= ^/3' + yi3'4-/3y' + y/ by the former. 
Hence the distributive principle is true in the multiplication of 

It only remains to show that it is true as to the scalar and 
vector parts of a quaternion^ and then we shall easily attain the 
general proof. 

Now, if a be any scalar, a any vector, and y any quaternion, 
[a'\'6)q = aj' + aj. 
For, if /3 be the vector in which the plane of q is intersected 
by a plane perpendicular to a, we can find other two vectors, - 
y and 5, in these planes such that 

y ^ 

And, of course, a may be written — ; so that 

^ y /3 y /3 

= aq-\- aq. 
And the conjugate may be written 

/(«'4-a') = /a'-f/a (§ 55). 
Hence, generally, 

(«-|-a)(^-fi3) = ab + afi-^-ba-^-afi: 
or, breaking up a and b each into the sum of two scalars, and 
a, fi each into the sum of two vectors, 

(fli -f «a -f ai -ha2)(^i -h^a -f i3i -f iQa) 

= K + «2)(<^i+^2) + («i-f«2)(/3i+i8,) + (*i-f*0(«i+aO 

+ (ai-fa,)(/3i+i3, 


(by what precedes, all the factors on the right are distributive, 
so that we may easily put it in the form) 

+ («a + aO(ia+^3). 
Putting «i +ai = py a^'^a^= q, h^ -f-^i = r, h^ +^a = s, 

we have 

(jo + ^)(r + *) =jor+^* 4 ?/• + ?«. 

82. For variety, we shall now for a time forsake the geo- 
metrical mode of proof we have hitherto adopted, and deduce 
some of our next steps from the analytical expression for a qua- 
ternion given in § 80, and the properties of a rectangular system 
of unit- vectors as in § 7 1 . 

We will commence by proving the result of § 77 anew. 

83. Let 

Then, because by § 71 every product or quotient of i^j, k is 
reducible to one of them or to a number, we are entitled to 

where w, ^, r], fare numbers. This is the proposition of § 80. 

84. But it may be interesting to find w, £, t;, f in terms of 
^j y, Zy x\ /, /. 

We have 


as we easily see by the expressions for the powers and products 
of i, j, k, given in § 7 1 . But the student must pay particular 


attention to the order of the factors^ else be is certain to make 
This (§ 80) resolves itself into the four equations 

/= <az + (^ —rix. 
The three last equations g^ve 

which determines <a. 

Also we have^ &om the same three^ 

which, combined with the first, gives 

y / — zy zx — xz xy — yx ' 

and the common value of these three fractions is then easily 
seen to be . 

x^-^y^ + z^ 
It is easy enough to interpret these expressions by means of 
ordinary coordinate geometry : but a much simpler process will 
be furnished by quaternions themselves in the next chapter, and, 
in giving it, we shall refer back to this section. 

85. The associative law of multiplication is now to be proved 
by means of the distributive (§ 81). We leave the proof to the 
student. He has merely to multiply together the factors 

W'\-xi-{'yj'\-zk, w'+x'i-\-yj-\-sfk, and v/^ + iiif'i-\-2^'j-\'/'k, 

as follows : — 

First, multiply the third factor by the second, and then 
multiply the product by the firit; next, multiply the second 
factor by the first and employ the product to multiply the third : 


always remembering that the multiplier in any product is placed 
before the multiplicand. He will find the scalar parts and the 
coefficients of f, /, hy in these products, respectively equal, each 
to each. 

86. With the same expressions for a, j3, as in section 83, 
we have 

= ^{xx'-\-yy 4- ^/) + (y^— ^/)i+ {zx'^xz)j-\- (xtf-yx") h. 

But we have also 

/3a = — (a?ar'+^/H-^/)— (y/— ^/)i-(2r/-;p/)y--(^— yarO*- 
The only difference is in the %ign of the vector parts. 

Hence ' /Sa)3 = /S/3a, (1) 

ra/3=-r/3a, (2) 

also aP+fia= 2 Safi, (3) 

aP-fia = 2raA (4) 

and, finally, by § 79, . 

afi=Kfia (5) 

87. If a = /3 we have of course (§ 25) 

x = x', y=/, z = /, 
and the formulae of last section become 

which was anticipated in § 73, where we proved the formula 

and also, to a certain extent, in § 25. 

88. Now let q and r be any quaternions, then 

Sqr = S.{8q+ Fq){8r^ Fr), 

= S.{8qSr + Sr.Fq+Sq,rr+ Fq Fr), 
^SqSr + S.FqFr, 
since the two middle terms are vectors. 


Similarly, Srq = 8r8q-\- 8, Fr Vq. 

Hence, since by (1) of § 86 we have 

8.rqrr ^ 8.rrrq, 

we see that 

Sqr=z 8rq, , (1) 

a formula of considerable importance. 

It may easily be extended to any number of quaternions, 
because, r being arbitrary, we may put for it rs. Thus we have 
(putting a dot after the 8 to shew that it refers to the whole 
product that follows it) 

8,qrs = 8.r8q 
= 8.8qr 
by a second application of the process. In words, we have the 
theorem — tke scalar of the product of any number of given qua- 
ternions depends only upon the cyclical order in which they are 

89. An important case is that of three factors, each a vector. 
The formula then becomes 

8.afiy = /S./Sya = /S.ya/3. 
But 8.aPy = 8.a{8py+ F^y) 

=1 8,a Fpyy since a8fiy is a vector, 
= ^8.aFyPy by (2) of § 86, 
= ^8.a{8yfi+Fyfi) 

Hence the scalar of the prodtcct of three vectors changes sign when 
the cyclical order is altered. 

Other curious propositions connected with this will be given 
later, as we wish to devote this chapter to the production of the 
fundamental formulae in as compact a form as possible. 

90. By (4) of § 86, 

Hence 2 F.aF^y = F.a {fiy-yP) 


(by multiplying both by a, and taking the vector parts of each 

(by introducing the null term ^ay—^ay). 
That is = jr(a/3H-)9a)y-r.()8/Say + ^ray + /S'ay./3H- Vay.ff) 
= r.{28aP)y-2r.fi8ay 
(if we notice that KFay.^^- ^.^^ay, by (2) of § 86). 

Hence F.aFfiy = ySafi^fiSya, (1) 

a formula of constant occurrence. 

Adding aSpy to both sides we get another most valuable 


Kafiy = aSPy-pSya + ySaP; (2) 

and the form of this shews that we may interchange y and a 
without altering the right-hand member. This gives 


a formula which may be greatly extended. 

91. We have also 

F, Fa^Fyb = - r. FybFa^ by (2) of § 86 : 

= bS.yFa^-yS.bFafi = b S.a^y^yS.afib, 


all of these being arrived at by the help of § 90 (1) and of § 89 ; 
and by treating alternately Fafi and Fyb as simple vectors. 
Equating two of these values, we have 

bS.afiy = aS,Pyb + pS.yab-\-yS.a^by (3) 

a very useful formula, expressing any vector whatever in terms 
of three given vectors. 

92. That such an expression is possible we knew already by 
§ 23. For variety we may seek another expression of a similar 


character^ by a process which differs entirely from that employed 
in last section. 

a, p, Y being any three vectors, we may derive from them 
three others Fafi, Ffiy, Fya\ and, as these will not generally 
be coplanar, any other vector h may be expressed as the sum 
of the three, each multiplied by some scalar (§ 23). It is re- 
quired to find this expression for h. 

Let 5 = a? Fafi ^yF^y + z Fya. 

Then Syb = xS.ya^ = xSafiy, 

the terms in y and z going out, because 

S.yFpy = S.yfiy = S0y^ = y^Sfi = 0, 

for y2 is (§ 73) a number. 

Similarly Sfib ^zS, Pya = zS. afiy, 

and SaJb = ^S.afiy. 

Thus bS. afiy = Fa^Syb + FfiySab + FyaSpb (4) 

93. We conclude the chapter by showing (as promised in 
§ 64) that the assumption that the product of two parallel 
vectors is a number, and that of two perpendicular vectors a 
third vector perpendicular to both, is not only useful and 
convenient but absolutely inevitable if our system, is to deal 
indifferently with all directions in space. We abridge Hamil- 
ton's reasoning. 

Suppose that there is no direction in space pre-eminent, 
and that the product of two vectors is something which has 
quantity, so as to vary in amount if the factors are changed, 
and to have its sign changed if that of one of them is reversed ; 
if the vectors be parallel, their product cannot be, in whole 
or in part, a vector inclined to them, for there is nothing to 
determine the direction in which it must lie. It cannot be 
a vector parallel to them ; for by changing the sign of both 
factors the product is unchanged, whereas, as the whole system 


has been reversed, the product vector ought to have been 
reversed. Hence it must be a number. Again, the product 
of two perpendicular vectors cannot be wholly or partly a 
number, because on inverting one of. them the sign of that 
number ought to change ; but inverting one of them is simply 
equivalent to a rotation through two right angles about the 
other, and (from the symmetry of space) ought to leave the 
number unchanged. Hence the product of two perpendicular 
vectors must be a vector, and an easy extension of the same 
reasoning shows that it must be perpendicular to each of the 
factors. It is easy to carry this farther, but enough has been 
said to show the character of the reasoning. 


1. It is obvious from the properties of polar triangles that 
any mode of representing versors by the sides of a triangle must 
have an equivalent statement in which they are represented by 
angles in the polar triangle. 

Show directly that the product of two versors represented 
by two angles of a spherical triangle is a third versor repre- 
sented by the stt;pple?/ient of the remaining angle of the triangle ; 
and determine the rule which connects the directions in which 
these angles are to be measured. 

2. Hence derive another proof that we have not genei^ally 

pq = qp. 

3. Hence show that the proof of the associative principle, 
§ 67, may be made to depend upon the fact that if from any 
point of the sphere tangent arcs be drawn to a spherical conic, 



and also arcs to the foci, the inclination of either tangent arc 
to one of the focal arcs is equal to that of the other tangent arc 
to the other focal arc. 

4. Prove the formulae 

2 S.aPy = apy--yPa, 
2r.a/3y= afiy+ypa. 

5. Show that, whatever odd number of vectors be represented 
^y tt, )3, y, &c., we have always 

V.afiyU = F.€bypa, 
F.aPybfCv = FrjCfbyfia, &c. 

6. Show that 

STapFpyFya =-{S.aPy)\ 

F.FafiF^yFya= Fa^iy^SaP-SfiySya)-^ , 

and F.{FapF.FfiyFya) = {pSay-aSfiy)S.aPy,- 

7. If a, p, y be any vectors at right angles to each other, 
show that 

(a>-f /3» + y»)/S.aj3y = a*Ffiy+fi*Fya + y*Fap. 

8. If a, )3, y be non-coplanar vectors, find the relations among 
the six scalars, a?, y, z and f, »y, f, which are implied in the 

xa^-y^^-zy = f r^y + TyFyaH-fra^. 

9. If a, )3, y be any three non-coplanar vectors, express any 
fourth vector, 5, as a linear function of each of the following 
sets of three derived vectors, 

F.ya^, r.a^y, F.^ya, 
and F.FapFpyFya, FF^yFyaFafi, FFyaFa^F^y. 

10. Eliminate p from the equations 

Sap = <?,' S^p = b, Syp = c, Sbp = d, 
where a, ^, y, 5 .are vectors, and a, b, c, d scalars. 



94. 4 MONG the most useful characteristics of the Calculus 
■^^ of Quaternions the ease of interpreting its formulse 

geometrically, and the extraordinary variety of transformations 
of which the simplest expressions are susceptible, deserve a 
prominent place. We devote this Chapter to the more simple 
of these, together with a few of somewhat more complex 
character but of constant occurrence in geometrical aid physical 
investigations. Others will appear in every succeeding Chapter. 
It is here, perhaps, that the student is likely to feel most 
strongly the peculiar difficulties of the new Calculus. But on 
that very account he should endeavour to master them, for the 
variety of forms which any one formula may assume, though 
puzzling to the beginner, is of the most extraordinary advantage 
to the advanced student, not alone as aiding him in the solution 
of complex questions, but as affi)rding an invaluable mental 

95. If we refer again to the figure of § 77 we see that 

0C= OBqosAOB, ' 
CB^OB^mAOB. ^ 

Hence, it 62 = a, OB = p, and Z AOB = 0, we have 
OB = Tfi, OA = Ta, 

OC = Tp cos^, CB = Tl3 sinO. 


„ ^p OC Tp ^ 

Hence 8- =. -p—- = -^ cos^. 

a UA Ta 

Similarly TF^ = -^ = ^sind?. 

Hence, if c be a unit- vector perpendicular to a and /3, or 

UOA a 

we have V- = -^ sin^ e. 

a 2a 

96. In the same way we may shew that 

and ra)3= TaT^smBri 

where t^ = UVafi = - iZF- . 


Thus ^>i^ *ca/tfr of the product of two vectors is the continued 
product of their tensors and of the cosine of the supplement of the 
contained angle. 

The tensor of the vector of the product of two vectors is the con- 
tinued product of their tensors and the sine of the contained angle; 
and the versor of the same is a unit^-vector perpendicular to both, 
and such that the rotation about it from the. first vector to the second 
is right-handed or negative. 

Hence TVafi is double the area of the triangle two of whose sides 
are a, /3. 


(a.) In any triangle ABC we have 

' AC=AB + SC. 

Hence AC' = 8 AC AC = 8. AC {IS -f £C). 

With the usual notation for a plane triangle the interpretation 
of this formula is 


— i'* = —be COB A -^ab cos C, 
or b =z a cos C+{?cos A. 

(6,) Again we have, obviously, 

or cb sin A = ca sin By 

, sin ^ sin B sin C 
whence = = . 

a b c 

These are truths, but not truisms, as we might have been led 
to fancy from the excessive simplicity of the process employed. 

98. Prom § 96 it follows that, if a and ^ be both actual 
(i. e. non-evanescent) vectors, the equation 

8,afi = 

shews that cos^ = 0, or that a is perpendicular to /3. And, in 
fact, we know already that the product of two perpendicular 
vectors is a vector. 
Again, if Va^^% 

we must have sin ^ = 0, or a u parallel to /3. We know already 

that the product of two parallel vectors is a scalar. 

Hence we see that « o r^ 
o ap = u 

is equivalent to a = Fyj3, 

where y is an undetermined vector ; and that 

is equivalent to a = xp, 

where x is an undetermined scalar. 

99. If we write, as in § 83, 

a = ix +jy -^kz, 
/3 = ix'^-jy'^-My 

64 QUATBRNlONa [chap. III. 

we have, at once, by § 86, 

Safi =: — a-y — yy'— zz' 

\ r r r r r r ^ 
where r = y/x^ +^' + 2;% / = y/» +/* + /*. 

Also Fa^ = „•'{ i:f-=^ i+ ^■Z^j+ ^^i}. 

^ rr rr ^ rr ^ 

These express in Cartesian coordinates the propositions we have 
just proved. In commencing the subject it may perhaps assist 
the student to see these more famiUar forms for the quaternion 
expressions ; and he will doubtless be induced by their appear- 
ance to prosecute the subject, since he cannot fail even at this 
stage to see how much more simple the quaternion expressions 
are than those to which he has been accustomed. 

100. The expression « ^ 

may be written S.{Va^)y 

because the quaternion aj3y may be broken up into 

{8ap)y + {raP)y 
of which the first term is a vector. 
But, by § 96, 

5.(ra/3)y = TaT^QmeSriy. 

Here Trj =^ l, let <t> be the angle between ri and y, then finally 

S.afiy --TaTfiTy sin cos <^. 
But as ly is perpendicular to a and /3, Ty cos <[> is the length of 
the perpendicular from the extremity of y upon the plane of 
a, p. And as the product of the other three factors is (§ 96) the 
area of the parallelogram two of whose sides are a, j8, we see that 
the magnitude of S.afiy, independent of its sign, is tAe volume of 
the parallelepiped of which three coordinate edges are a, )3, y; or 
six times the volume of the pyramid which has a, )3, y for 


IQL Hence the equation 

S.a^y = 0, 
if we suppose a, ^, y to be actual vectors, shews either that 
sin fl = 0, 
or cos </) = 0, 

i. e. two of the three vectors are jparallel, or all three lie in one 

This is consistent with previous results, for if y = pfi we have 
S.aPy=p8.aP^ = 0; 
and, if y be coplanar with a, p, we have y ^pa-^qfi, and 
S.apy = S.aPipa-^qP) = O'. 

102. This property of the expression S.a^y prepares us to find 
that it is a determinant. And, in &ct, if we take a, ^ as in 
§ 83, and in addition 

y = iy'+y/'+*/', 
we have at once 

X y z 
x' / / 

The determinant changes sign if we make any two rows change 
places. This is the proposition we met with before (§ 89) in 
the form 

S.Q^y = -^8. Pay = S.fiya, Sec. 

If we take three new vectors 

Oi = ix-^jV-^h/^y 

yi =i^-|-y/-|-*/', 
we thus see that they are coplanar if a, /3, y are so. That is, if 
S.afiy = 0, 
then &aj/3iyj = 0. 



103. We hsve, bj § 52. 

(Tqy « qKq = (8i+Fq)(8q-Fq) (§ 79), 
= (Sqy-iFqy by algebra, 
= {8qy+(TFqy (§ 73). 
If ^ = afij we have Kq = pa, and the formula becomes 

ap.fia = a«/3« = (Saj8)«-(rai3)». 
In Cartesian coordinates this, is 

More generally we have 

= jf Zf JTj (§ 56) = (r?)' (2V)» (§ 62). 
If we write 

jf = «? +a = w +ip +^y +fe, 

this becomes 

(w^+x^ +y» + -2;»)(«?'» +a:'» +/« -h/^) 

-h («?/ -h «? y -h ;Ka?' — or/)* + («?/ -h «p'2J + ir/ — ^/)', 
a formula of algebra due to Euler. 

104« We have^ of course, by multiplication, 

(a+/3)« = a» + ai9-l-/3a + 0» = a»-h2Sa/3 + 0» (§86(3)). 
Translating into the usual notation of plane trigonometry, this 
becomes c» = a» - 2 a* cos C+ 5*, 

the common formula. 

Again, ir(a+/3)(a-i8) = - ra)3+ V^a = -2 Fa/S (§ 86 (2)). 

Taking tensors of both sides we have the theoren\, tie paral- 
lelogram wAose sides are parallel and equal to the diagonals of a 
given parallelogram, has dovile its area (§ 96). 


Also iS(a+j3)(a-/3) = o»-/3% 

and vanishes only when a^ = /3», or Ta^ Tfi; that is, Ue 
diagonals of a parallelogram are at right angles to one another y 
when, and only wheUy it is a rhombus. 

Later it will be shewn that this contains a proof that the 
angle in a semicircle is a right angle. 

105. The expression 

p = aj9a~* 

obviously denotes a vector whose tensor is equal to that of )3. 

But we have ^.^,^ = 0, 

so that p is in the plane of a, )3. 

Also we have 

Sap = Saft 

so that )3 and p make equal angles with a, evidently on opposite 
sides of it. Thus if a be the perpendicular to a reflecting surface 
and /3 the path of an incident ray, p will be the path of the 
reflected ray. 

Another mode of obtaining these results is to expand the 
above expression, thus, § 90 (2) 

p = 2a-*/Sa)3-i9 

= a-H^ai3-ira/3), 

so that in the figure of § 77 we see that if OA = a, and OB = /3, 
we have OB = p = a/3a"^ 

106. For any three coplanar vectors the expression 

p = a^y 
is (§ 101) a vector. It is interesting to determine what this 
vector is. The reader will easily see that if a circle be described 
about the triangle, two of whose sides are (in order) a and )3, 
and if from the extremity of j3 a line parallel to y be drawn 

K % 


again cutting the circle^ the vector joining the point of inter- 
section with the origin of a is the direction of the vector apy. 
For we may write it in the form 

p = a^«/3- V = -{Tpy ap-^y = -(I]3)» | y, 

which shows that the veraor which turns fi into a direction 
parallel to a, turns y into a direction parallel to p. And this 
is the long known property of opposite angles of a quadrilateral 
inscribed in a circle. 

Hence if a, /3, y be the sides of a triangle taken in order, the 
tangents to the circumscribing circle at the angles of the 
triangle are parallel respectively to 

a/3y, Pya, and yafi. 

Suppose two of these to be parallel, i. e. let 

aPy=xPya=xayP (§ 90), 

since the expression is a vector. Hence 

which requires either 

a?=l, ry^=0, or y||/3, 

a case not contemplated in the problem ; 

or a? = — 1, SPy=0, 

i. e. the triangle is right-angled. And geometry shews us at 
once that this is correct. 

Again, if the triangle be isosceles, the tangent at the vertex 
is parallel to the base. Here we have 


or x{a -h y) = a{a -h y)y ; 

whence ir=y" = a^, or Ty=:Ta, as required. 

As an elegant extension of this proposition the reader may 
prove that the vector of the continued product afiyb of the 
vector-sides of a quadrilateral inscribed in a sphere is parallel to 
the radius drawn to the comer (a, 6). 


107. To exemplify the variety of possible transformations 
even of simple expressions, we will take two cases which are 
of frequent occurrence in applications to geometry. 

Thus TO) + a)=T(p-a), 

[which expresses that if 

Q2=a, 6jL--a, and QP=p, 
we have AP^A'P 

and thus that P is any point equidistant from two fixed points,] 

may be written 

(p + a)«=(p-a)% 

or p« + 2/Sap + a> = p« — 2iSap + a» (§ 104), 
whence Sap=0. 

This may be changed to 

ap-h pa=0, 
or ap + Jrap=0, 


or finally, TVU^=zl, 

aU of which express properties of a plane. 

Again, Tp-Ta 

may be written ^p _ 


(p-ha)»-2/Sa(p + a) = 0, 

p = (p-i-a)-*a(p-|-a), 

'8'(p + a)(p— a)=0, or finally, 

7.(p + a)(p-a) = 2Trap. 

All of these express properties of a sphere. They will be 
interpreted when we come to geometrical applications. 


108. We have seen in § 95 that a quaternion may be 
divided into its scalar and vector parts as follows : — 

^ = 5^+ F^ = -^ (cosO + esin^r; 
a a a la 

where is the angle between the directions of a and p, and 


€ = UV- is the unit-vector perpendicular to the plane of a 

and j3 so situated that positive (i. e. left-handed) rotation about 
it turns a towards /3. 
Similarly we have (§ 96) 

ap = 8afi+Vap 

= Ta2)3 (— cos fl— € sin 0), 
$ and € having the same signification as before. 

109. Hence, considering the versor parts alone, we have 

£/■- = — Uafi = cos ^ + € sin ^. 

Similarly t/^ = cos</)-f €sin^; 

</> being the positive angle between the directions of y and J3, 
and € the same vector as before, if a, ^, y be coplanar. 
Also we have 

U- = cos (^ + <^) + c sin {0+<t)). 

But we have always 

^ . - = ~ , and therefore 
pa a' 

/3 a a 

or cos(0+^)-f €sin(<^4-^) = (cos^ + €sin<^)(cos^+esin^) 

= cos ^ cos fl— sin (^ sin ^ + € (sin </> cos d+ cos </> sin 0), 
from which we have at once the fundamental formulae for the 


cosine and sine of the sum of two arcs^ hj equating separately 
the scalar and vector parts of these quaternions. 

And we see^ as an immediate consequence of the expressions 
above, that 

cos m$'^€ sin mO = (cos 0+ c sin d)** 

if w be a positive whole number. For the left-hand side is a 
versor which turns through the angle mO 2it once, while the 
right-hand side is a versor which effects the same object by m 
successive turnings each through an angle 6. See § 8. 

UO. To extend this proposition to fractional indices we have 

only to write - for 6, when we obtain the results as in ordinary 


Prom De Moivre's Theorem, thus proved, we may of course 
deduce the rest of Analytical Trigonometry. And as we have 
already deduced, as interpretations of self-evident quaternion 
transformations (§§ 97, 104), the fundamental formulsB for the 
solution of plane triangles, we will now pass to the consideration 
of spherical trigonometry, a subject specially adapted for treat- 
ment by quaternions ; but to which we cannot afford more than 
a very few sections. The reader is referred to Hamilton's works 
for the treatment of this subject by quaternion exponentials. 

HI. Let a, /3, y be unit- vectors drawn from the centre to the 
comers A, -B, C of a triangle on the unit-sphere. Then it is 
evident that, with the usual notation, we have (§ 96), 

Sap ss —cos c, Sfiy = —cos a, 8ya = —COS dy 

TVafi= sine?, TVpy ^ sin a, TVya= smb. 

Also UVafi^ UVfiy, UVya are evidently the vectors of the corners 
of the polar triangle. 

Hence S. UFap UVpy = cos £, &c., 

TV. UVap UVfiy = sin B, &c. 


Now (§ 90 (1)) we have 

SVapVpy = S.aV.prpy 

Remembering that we have 

SFaprpy = TFapTVPySMFapUrpy, 
we see that the formula just written is equivalent to 
sinasin^roosJSs:— cosaco8(?+cosdj 
or co8& = co8acos(?+sina8in(?co8^. 

112« Again^ 

which gives 

TF.FapF^ = S.aPy = S.aFPy = S.^Fya = S.yVafi, 
or sinasint^sinJS = sinasinj?^ s sin^sinjo^ = sint^sinj^^; 
where j9a is the arc drawn from A perpendicular to BC, &c. 

Hence ^^fia = sin <; sin B, 

sinasinc . _, 
sin 19b = — ' — 2 — sm /!• 

smp^ = sin a sin B. 

113. Combining the results of the last two sections^ we have 
Fafi. Fpy = sin a sin <; cos £ — j3 sin a sin csiaB 
= sin a sin <; (cos JS^ j3 sin JS). 
Hence U. FaB Fpy = (cos J8 - j8 sin B), ] 

and U.FyPFpa = (cos B+fimn B). 

These are therefore versors which turn the system negatively or 
positively about OB through the angle B. 
As another instance^ we have 


^-^ =&c 



The interpretation of each of these forms gives a different 
theorem in spherical trigonometry. 

114. A curious proposition, due to Hamilton, gives us a qua- 
ternion expression for the spherical excess in any triangle. The 
following proof, which is very nearly the same as one of his, 
though by no means the simplest that can be given, is chosen 
here because it incidentally gives a good deal of other in- 
formation. We leave the quaternion proof as an exercise. 

Let the unit-vectors drawn from the centre of the sphere 
to Ay By C, respectively, be a, /3, y. It is required to express, 
as an arc and as an angle on the sphere, the quaternion 

The figure represents an orthographic projection made on a 
plane perpendicular to y. Hence C is the centre of the circle 
BEe. Let the great circle through Ay B meet BEe in Ey Cy 
and let BE be a quadrant. Thus BE represents y (§ 72). Also 
make EF=uiB== pa-\ Then, evidently, 

which gives the arcual representation required. 


Let BF cut Ee'm 0. Make Ca =r EQj and join B^ a, and 
a, F. Obviously^ as 2> is the pole of Ee, fiaiBBi quadrant ; and 
since FO = Ca, Ga ^ FC, a quadrant also. Hence a is the 
pole of DO, and therefore the quaternion may be represented 
by the angle DaF. 

Make Cb^ Ca, and draw the arcs Pafi, Pha from P, the pole 
of AB. Comparing the triangles Fba and ea^, we see that 
Fa = e^. But, since P is the pole of AB, F^a is a right angle : 
and therefore as Pa is a quadrant, so is F^. Thus AB is the 
complement of Fa or fie,, and therefore 


Join bA and produce it to c so that Ac =^bA; join c, P, 
cutting AB in o. Also join c, B, and j5, a. 

Since P is the pole of AB, the angles at o are right angles ; 
and therefore, by the equal triangles baA, coA, we have 

aA = Ao. 

But ai3 = 2AB, 

whenise oB = Bfi, 

and therefore the triangles coB and JSa^ are equal, and e, B, a 
lie on the same great circle. 

Produce cA and cB to meet in H (on the opposite side of the 
sphere). H and c are diametrically opposite, and therefore cP, 
produced, passes through H. 

Now Pa = Pb = PR, for they di£Per from quadrants by the 
equal arcs ap, ba, oc. Hence these arcs divide the triangle Eab 
into three isosceles triangles. 

But LPEb^LPHa = LaHb = Lbca. 

Also L Pab = ff- Lcab-L PaH, 

Z Pba = LPab = tt-Z cba-LPbH. 

Adding, 2 ZPai = It:—- Lcab-- Lcba — Lbca 
= -JT— (spherical excess of adc). 


But^ as Z Fa^ and Z Dae are right angles^ we have 
angleof/3o~*y = LFaJD^ L^aer=, LPab 

= - — ^ (spherical excess oiabc). 

[Numerous singular geometrical theorems, easily proved ab 
initio by quaternions, follow from this : e. g. The arc AB, 
which bisects two sides of a spherical triangle abc^ intersects 
the base at the distance of a quadrant from its middle point. 
All spherical triangles, with a common side, and having their 
other sides bisected by the same great circle (i. e. having their 
vertices in a small circle parallel to this great circle) have equal 
areas, &c., &c.] 

115. Let 0« = a', Oi = /S', 0^ = /, atid we have 

= Ca.BA 

=: JEG.F^ =^ f^. 

But FG is the complement of DF, Hence the an^le of the 


is half the spherical excess of the triangle whose angula/r points are 
at the extremities of the unit^vectors o', ^, y\ 

[In seeking a purely quaternion proof of the preceding pro- 
positions, the student may commence by showing that for any 
three unit- vectors we have 

a ^ y ^ '* 

The angle of the first of these quaternions can be easily assigned ; 
and the equation shows how to find that of fia^^y. But a still 
simpler method of proof is easily derived from the composition 
of rotations.] 

L 2 

76 QUATERNIONS. [chap. IH. 

118. A scalar equation in p^ the vector of an undetermined 
pointy is generally the equation of a surf ace; since we may 
substitute for p the expression 

where or is an unknown scalar^ and a any assumed unit- vector. 
The result is an equation to determine x. Thus one or more 
points are found on the vector xa whose coordinates satisfy the 
equation ; and the locus is a surface whose degree is determined 
by that of the equation which gives the values of x. 

But .a vector equation in p^ as we have seen^ generally leads 
to three scalar equations^ from which the three rectangular or 
other components of the sought vector are to be derived. Such 
a vector equation^ then^ usually belongs to a definite number of 
jmnts in space. But in certain cases these may form a line, and 
even a surface, the vector equation losing as it were one or two 
of the three scalar equations to which it is usually equivalent. 

Thus while the equation 

gives at once 

which is the vector of a definite point (since we have evidently 

the closely allied equation 

is easily seen to involve 

and to be satisfied by 

whatever be x. Hence the vector of any point whatever in the 
line drawn parallel to a from the extremity of a"*^ satisfies the 
given equation. 

117. Again, FapTp^^iFafiy 

is equivalent to but two scalar equations. For it shews that 


Vap and Ffip are parallel^ i. e. p lies in the same plane as a and p, 
and can therefore be written (§ 24) 

where x and y are scalars as yet undetermined. 

We have now 

Fop =yrap, 


which, by the given equation, lead to 

dy=l, or y=-, or finally 


1 ^ 


which (§ 40) is the equation of a hyperbola whose asymptotes 
are in the directions of a and ft. 

US. Again, the equation 


though apparently equivalent to three scalar equations, is really 

equivalent to one only. In fact we see by § 91 that it may 

be written 

— ajS.aj9p=0, 

whence, if a be not zero, we have 
and thus (§ 101) the only condition is that p is coplanar with 
a, p. Hence the equation represents the plane in which a and 

lift Some very curious results are obtained when we extend 
these processes of interpretation to functions of a quaternion 

instead of functions of a mere veetor p. 

A scalar equation containing such a quaternion, along with 
quaternion constants, gives, as in last section, the equation of 


a euiface, if we assign a definite value to «. Hence for succes- 
sive values of », we have successive surfaces belonging to a 
system ; and thus when o is indeterminate the equation repre- 
sents not a surface, as before^ but a volume, in the sense that the 
vector of any point within that volume satisfies the equation. 
Thus the equation (TV—a^ 

or ©•— p*=a*, 

or (TpY = a»-a)», 

represents^ for any assigned value of q^ not greater than a, a 
sphere whose radius is \/a*— «•. Hence the equation is satisfied 
by the vector of any point whatever in the volume of a sphere 
of radius a, whose centre is origin. 

AgaiUj by the same kind of investigation^ 

where ;=:a> -hp is easily seen to represent the volume of a sphere 
of radius a described about the extremity of /3 as centre. 

Also 8{q*)-=z —a' is the equation of infinite space less the space 
contained in a sphere of radius a about the origin. 

Similar consequences as to the interpretation of vector equa- 
tions in quaternions may be readily deduced by the reader. 

120. The following transformation is enunciated by Hamilton 
{Lectures, p. 587, and JElements, p. 299). 

r-^(r*j»)*^» = U{rq+KrKq). 
Let r-*(r» j»)*y-» = t, then 

^ = Ij and therefore 
Kt = ^* ; 

But (r»?»)* = rtq, 

or r^q^ = rtqrtq, 

or rq^tqrt,. 

Hence KqKr = tr^KrKqt-\ 

or KrKq = tKqKrt. 


Thus we have 

U{rq±KrKq) = tU{qr±KqKr)t, 
or, if we put « = V'{qr±KqKr), 

Ks = ±tst. 
Hence sKs = (Ts)' = 1 = ± stst, 

which, if we take the positive sign, requires 

*^ = ± 1, 
or t=: ±8-^ = ± UKs, 

which is the required transformation. 

[It is to be noticed that there are other results which might 
have been arrived at by using the negative sign above ; some 
involving an arbitrary unit-vector, others involving the ima- 
ginary of ordinary algebra.] 

12L As a final example, we take a transformation of Hamil- 
ton's, of great importance in the theory of surfaces of the 
second order. 

Transform the expression 

in which a, )3, y are any three mutually rectangular vectors, 
into the form 

. y(ip+pK) x' 

which involves only two vector-constants, *, k. 

{T(ip+pK)}» = (ip+p^)(pi + ^p) (§§52,55) 

= (l* + K*)p' H- {l^pKp + p#Cpi) 

= (i» + ic«)p» + 2&ipicp 
= (i-jc)V + 4&p%. 

^ Hence (^ap)« + (5^p)- + (%)» = i^^ 

But a'^iSapy+fi-^iSM'+y-'iSypy = p' (§§ 25, 73). 

80 QUATEBNIONfl. [chap. III. 

Multiply by /3' and subtract^ we get 


The left side breaks up into two real factors if J3* be intermediate 
in value to a* and y* : and that the right side may do so the 
term in p* must vanish. This condition gives 

P* = . ,_ ,x, ; and the identity becomes 

5(av/(i-^) + yv^(7-i)M(«v/(i-|^)-y^(f!-i))p 

Hence we must have 

-^=^(av'(l-^) + yv/(^4-l)> 

-j_j = -(av/(i-^)-yv(^-i)> 

where jt7 is an undetermined scalar. 
To determine JD, substitute in the expression for p*, and we find 

= (i'*+i)(«*-y')-2(«*+y')+4/3'. 


Thus the transformation succeeds if 

1 / a^ 

which gives jd -f - = + 2 v/— 

JD "" a'— y* 

JO ~ " a" — y' 
Hence ^^^ = (^-i*') («'->') = ±4x/^', 
or (k»-i»)-»= +ya2V- 


Again, p= /-T— ^. > -= y - ^ 

and therefore 

Thus we have proved the possibility of the transformation, and 
determined the transforming vectors t, k. 

122. By differentiating the equation 

(&V>)* + (5^P)«+(W = (^±^)' 
we obtain, as will be seen in Chapter IV, the following, 

"where p' also may be any vector whatever. 

This is another very important formula of transformation; 
and it will be a good exercise for the student to prove its truth 
"by processes analogous to those in last section. We may merely 
observe, what indeed is obvious, that by putting p^= p it be- 
comes the formula of last section. And we see that we may 
write, with the recent values of i and k in terms of o, j3, y, the 

aSap + pSpp + ySyp = ^ (^Jj.^y 

^ {L-Kyp + 2(lSKp+KSip) 

123. In various quaternion investigations, especially in such 
as involve imaginary intersections of curves and surfaces, the old 
imaginary of algebra of course appears. But it is to be par- 
ticularly noticed that this expression is analogous to a scalar 
and not to a vector, and that like real scalars it is commutative 
in multiplication with all other factors. Thus it appears, by the 

82^^ QUATBRNI0N8. [cHAP. IE. 

same proof as in algebra, that any quaternion expression which 
contains this imaginary can always be broken up into the sum 
of two parts, one real, the other multiplied by the first power 
of \/— 1. Such an expression, viz. 

where / and /' are real quaternions, is called a biquatesnion. 
Some little care is requisite in the management of these ex- 
pressions, but there is no new diflSculty. The points to be 
observed are : first, that any biquatemion can be divided into 
a real and imaginary part, the latter being the product of 
'/—I by a real quaternion; second, that this \/— 1 is com- 
mutative with all other quantities in multiplication; third, that 
if two biquatemions be equal, as 

we have, as in algebra, 

80 that an equation between biquatemions involves in general 
eight equations between scalars. Compare § 80. 

124. We have, obviously, since \/— 1 is a scalar. 
Hence (§ 103) 

The only remark which need be made on such formulse is this, 
that the temor of a Hquatemion may vanish mhih both of the 
component quaternions are finite. 

Thus, if T/= Tf, 

and 84 K4'- 0, 

the above formula gives 

The condition S.^Kf- 

may be written 


Kf^^'^a, or /'=-aZ/-»=- 

where a is an indeterminate vector. 

Hence T^=Tf:=TKf=~y 

and therefore 

is the general form of a biquaternion whose tensor is zero. • 

125. More generally we have, q, r, /, / being any four real 
and non-evanescent quaternions. 

That this product may vanish we must have 
qr = //, 
and q/=i^^r. 

Eliminating / we have 

which gives (/"*?)* = ~ 1^ 

i. e. J = ^a 
where a is some unit- vector. 

And the two equations now agree in giving 
— r = a/, 
so that we have the biquaternion factors in the form 

^{a-{- sT^) and -(a- V^)/ ; 
and their product is 

-^(a+ y^) (a- V^)/, 
which, of course, vanishes. 

[A somewhat simpler investigation of the same proposition 
may be obtained by writing the biquatemions as 

M % 

84 QUATBRNIOKa [chap, ni- 

or /(/'+yZT) and (/'+>/^)/, 

and showing that 

j^'=: -/'=o, where Ta = 1.] 

From this it appears that if the product of two Hvectors 
p+o-^/— 1 and p'+ </\/^— 1 
is zero^ we must have 

where a may be any vector whatever. But this result is still 
more easily obtained by means of a direct process. 

126. It may be well to observe here (as we intend to avail 
ourselves of it in the succeeding Chapters) that certain ab- 
breviated forms of expression may be used when they are not 
liable to confose^ or lead to error. Thus we may write 

T^q for {TqY, 
just as we write 

cos*^ for (cos^)S 

although the true meanings of these expressions are. 
T{Ta) and cos (cos ^). 

The former is justifiable, as T{Ta) = Ta, and therefore T^a is 
not required to signify the second tensor (or tensor of the 
tensor) of a. But the trigonometrical usage is quite inde- 

Similarly we may write 

S'q for {SqY, &c. 
but it may be advisable not to use 


as the equivalent of either of those just written ; inasmuch as 
it might be confounded with the (generally) different quantity 

8.q^ or S{q% 
although this is rarely written without the point or the brackets. 


127. The beginner may expect to be a little puzzled with 
this aspect of the notation at first; but, as he learns more of 
the subject, he will soon see clearly the distinction between such 
an expression as 

where we may omit at pleasure either the point or the first V 
without altering the value, and the very diflferent one 

which admits of no such chatiges, without altering its value. 

All these simplifications of notation are, in fact, merely ex- 
amples of the transformations of quaternion expressions to which 
part of this Chapter has been devoted. Thus, to take a very 
simple example, we easily see that 

S.FapFpy = SFa^py = S.apFfiy = SaF.^Ffiy^ ^SaF{Ffiy)p 

= 8aF{Fyp)P = 8.aF{yp)P = 8.F{yfi)fia = SFyfiFfia 

= S.ypFpa z= Sec, Sec. 

The above group does not nearly exhaust the list of even the 
simpler ways of expressing the given quantity. We recommend 
it to the careful study of the reader. 


1. Investigate, by quaternions, the requisite formulae for 
changing jfrom any set of coordinate axes to another; and 
derive from your general result, and also from special investi- 
gations, the common expressions for the following cases : — 

(a.) Rectangular axes turned about z through any angle. 


(b.) Bectangolar axes turned into any new position by 
rotation about a line equally inclined to the three. 

(c.) Bectangnlar turned to oblique, one of the new axes 
lying in each of the former coordinate planes. 

2. If Tp = It = 2)3= 1, and = 0, show by direct trans- 
formations that 

Interpret this theorem geometrically. 

3. If So^ = 0, 7a = 7/3 = 1, show that 

(1 +o*)j3 = 2cos ^<?p = 2 SJ.a^fi. 

V 4. Put in its simplest form the equation 

pS.raprpyVya = aF.ryaFaP + br.rafiFfiy + cr.r^rya; 

and show that a ^ o. 

a = S.pyp, &c. 

1/^. Prove the following theorems, and exhibit them as pro- 
perties of determinants : — 

(a.) &(a + i8)(/3H-y)(y + o) = 2&a^y. 

(«.) 8.Faprpyrya = -{8.aPyy. 

{c.) 8.F{a + P)(fi + y)F{fi-^y)(y + a)F{y + a){a + P)= -4(&a^y)^ 

{d.) S.F{FafiFPy)F{FfiyFya)F{FyaFaP)=-{8.afiyy. 

{e.) &8€f=-16(iS.a)3y)*, where 

j= r(r(a+i3)(/34-y)r(^ + y)(y+a)), 

€= r(r(i3 + y)(y + a)r(y-ha)(a + /3)), 
f = r(r(y + a)(a+^)r(a + /3)0+y)). 

6. Prove the common formula for the product of two de- 
terminants'^of the third order in the form 

S.apy8.aiPiyi = 

ScMi Sfiai 8yai 
Sap, SfiP, SyP, 
Say, Spy, 8yy, 


y 7. The lines bisecting pairs of opposite sides of a quadrilateral 

are perpendicular to each other when the diagonals of the qua- 
drilateral are equal. 

•^0 i. Show that 

(i.) 8.(f = S'qS SqT^ Fq, 

-— '(rf.) 8{F.apyF.fiyaF.yaP) = ^Sa^SpySyaS^afiy, 

(e.) F.q':=z{3 8'q-^T^Fq)Fq, 

(/.) qUFr^ ^Sq.UFq^TFq; 

and interpret each* as a formula in plane or spherical trigo- 

>' 9. If 9 be an undetermined quaternion^ what loci are repre- 
sented by 

{a.) {qa-'y^^a\ 

{b.) {qa-'y = a*, 

where a is any given scalar and a any given vector? 

y 9Z Sf^ If J be any quaternion^ show that the equation 

<2« = q' 
is satisfied, not alone by Q^±q but also, by 

Q = ± ^sf:^{8q.T}Fq-TFq\ 

(Hamilton, Lectures, p. 673.) 

,2 ^1 1- Wherein consists the difference between the two equations 

T^^^l, and (^'=-1? 

What is the full interpretation of each, a being a given, and 
p an undetermined, vector ? 


)U y 1 2. Knd the fall oonseqaenoes of each of the following groups 
of equations, both as regards the unknown vector p and the 
given vectors a, fi, y. 

SaSa^O Sap ^0, Sap =0, 

o.pyp^v, g^^ _^^ S.aPYp=0. 

, '', i 13. From §§ 74^ 109^ show that^ if c be any unit-vector, and 

many scalar, €** = cos— — hcsin-— • 

2 2 

Hence show that if a, j3, y be radii drawn to the comers of 
a triangle on the unit-sphere, whose spherical excess is m, right 
»°8rles, a+^ y+a ^+y ^ ^ 

/3 + y'a+i3'y + a 

Also that, i£Ag£, Che the angles of the triangle, we have 

tC tB iA 

y»j3»o»=;— 1. 

I i/14. Show that for any three vectors a, 0, y, we have 

{Uapy + {Ufiyy-{-{Uayy + {U.aPyy + 4: Uay.8UapSUfiy =-^2. 

(Hamilton, Elements, p. 388.) 

^"^ 1/16. If «!, a„ a„ X, be any four scalars, and pi, p„ p, any three 
vectors, show that 

(5.piP>p.)* + (2.«,rp,p.)«+a?«(2rp,p3)»-a?'(2.«,(p,~Pi))« 

+2n(a?»+5pip,+tf a) = 2n(a?«+v)+2na» /^ 

+ 2{(a:«+a,«+p,«)((rp.p,)« + 2a,«3(^« + %p.)--;p«(p.-p,)*)}; 

where Ua* = «i«a,«a,«. 
Verify this formula by a simple process in the particular case 

^1 = A, =r a, ST 07 = 0. 




128. TN Chapter I we have already considered as a special 
-■- case the differentiation of a vector function of a 
scalar independent variable : and it is easy to see at once that 
a similar process is applicable to a quaternion ftmction of a 
scalar independent variable. The differential, or differential 
coefficient, thus found, is in general another function of the 
same scalar variable ; and can therefore be differentiated anew 
by a second, third, &c. application of the same process. And 
precisely similar remarks apply to partial differentiation of a 
quaternion function of any number of scalar independent vari- 
ables. In fact, this process is identical with ordinary differ- 

129. But when we come to differentiate a function of a 
vector, or of a quaternion, some caution is requisite; there is, 
in general, nothing which can be called a differential coefficient; 
and in fact we require (as already hinted in § 36) to employ 
a definition of a differential, somewhat different from the or- 
dinary one, but coinciding with it when applied to functions 
of mere scalar variables. 

130. If r = F{q) be a ftinction of a quaternion q, 

dr=dFq=J^^n{F{q-\- ^)-F(q)h 


where n is a scalar which is ultimately to be made infinite^ is 
defined to be the differential of r or Fq, 

Here dq maj be any quaternion whatever, and the right-hand 
member may be written 

/(?. ^)> 
where /is a new function^ depending on the form oiF; homo- 
geneous and of the first degree in ^; but not^ in general^ 
capable of being put in the form 


13L To make more clear these last remarks, we may observe 
that the function ^, , x 

thus derived as the differential oiF{q)y is distributive with respect 

to dq. That is 


r and s being any quaternions. 

For Aq.r+») = J^n{F{q+'^-F{q)) 


And, as a particular case, it is obvious that if ;r be any scalar 
/(?, ar) = x/{q, r). 
132. And if we define in the same way 

dF(q, r,» ) as being the value of 

•C.«{^(?+ ?.»•+?,*+$. )-nq,r,s, )}, 

where q,r,s, ,.. dq,dr,dsy are any quaternions whatever ; 

we shall obviously arrive at a result which may be written 

Aq>r,s,..,dq,dr,ds, ), 


where /is homogeneous and linear in the system of quaternions 

dqydrydSj and distributive with respect to each of them. 

Thus, in differentiating any power, product, &c. of one or more 
quaternions, each factor is to be differentiated as if it alone 
wei:^ variable ; and the terms corresponding to these are to be 
added for the complete differential. This differs from the 
ordinary process of scalar differentiation solely in the feet that, 
on account of the non-commutative property of quaternion mul- 
tiplication, each factor must be differentiated in situ. Thus 

d{qr) = dq.r-\-qdrf but not generally = rdq-^qdr. 

133. As Examples we take chiefly those which lead to results 
of constant use to us in succeeding Chapters. Some of the 
work will be given at full length as an exercise in quaternion 

(1) (2'p)'=-p'. 

The differential of the left-hand side is simply, since 2}> is a 

«^^> 2TpdTp. 


= <.«(!*^/'+^)<5'<") 

^n n 

= 2 Spdp. 


TpdTp = —Spdp, 


dTp = -8.Updp=8 * , 


dTp dp 




dp = dTp.Up'^TpdUp, 

N 2, 


, dp dTp dVp 

whence -T. = 4. 

p Tp Up 

Hence dUp _ jr rfp 

This may be transfonned into 

jrdpp ^ &c. 

(3) {Tqy^^qKq 

2TqdTi = d(qKq) = J^^n{{q^.^)K{q-^^)-qKq^, 

= qKdq-^dqKq, 
=:2S.qKdq = 28.Kqdq. 
Hence rf^y = 8. UKqdq = S. I7y-»^ 

since Tq = My, and UKq = ?7f-^ 

If y = p, a vector, Kq=^Kp:=z-^p, and the formula becomes 
dTp^^S.Updp asin(l). 

But dq = Ty^E/y + f/y(?ry, 

, . , . dq dTq dUq 

which gives !^=^+-^; 

whence, as S — = ~^ , 
q Tq 

we have F— * = -fj^ • 

q Uq 


(4) ^(?')=c^((2+fy-?o 

= qdq + dqq 

= 2S.qdq-^2Sq.V€lq + 2Sdq.Fq. 
If ^ be a vector, os p, Sq and 8dq vanish, and we have 
d(p^) = 2Spdp as in (1). 

(5) Let q = /•*. 

This gives dr^ = dq. But 

dr = ^(y*) = qdq-^-dqq. 
This, multiplied Jy j and into Kq, gives 
ydr = q^dq-i-qdqq, 
and drKq:=r dqTq^-^qdqKq. 

Adding, we have 

qdr + (^rJCy = {q^ + Tj' + 2 Sjr. jr)rf^ ; 

whence dq, i. e. d/*, is at once found in terms of dr. This pro- 
cess is given by Hamilton, Lectures, p. 628. 

(6) qq-^ = 1, 
qdq-^-^dqq-^ = 0; 

.'. dq-^ =^^^dqq-^. 

If J is a vector, = p suppose, 

dp~^ =z—p-^dpp~^ 

_^_ 2^^ 
p^ p p 

^p p^p 

(7) q = Sq+rq, 

But dq=: Sdq-i-Fdq. 

94 QUATERNIONS. [chap. IV. 

Comparing, we have 

dSq = 8dq, dVq = Vdq. 
Since Kq=: 8q^ Fq, we find by a similar process 
dKq = Kdq. 

134. Saccessive differentiation of course presents no new 
Thus, we have seen that 

d{q*) zzzdqq-^-qdq. 
Differentiating again, we have 

d'{q') = d'q.q-i-2{dqy-^qd'q, 
and so on for higher orders. 

ISqhesL vector, as p, we have, § 133 (1), 

d{p^) = 28pdp. 
Hence d*{p*) = 2{dpy + 28pd*pj and so on. 

Similarly d'Up = ^d{^Vpdp). 


and d. Vpdp = V.pd'p. 

Bat rf-L = _i^ = £M', 


[This may be farther simplified; but it may be well to 
caution the student that we cannot, for such a purpose, write 
the above expression as 

^ V.p {dpVpdp^d-pp- + 2dpSpdp}.} 

135. If the first differential of q be considered as a constant 
quaternion^ we have, of course, 

d^q = 0, d^q = 0, &c., 
and the preceding formulae become considerably simplified. 


Hamilton has shown that in this case Taylor^s Theorem admits 
of an easy extension to quaternions. That is^ we may write 

Aq+xdq) =/iq)+sd/(q)+ ^d^Aq) + 

if d*q=0; subject, of course, to particular exceptions and limita- 
tions as in the ordinary applications to fiinctions of scalar 
variables. Thus, let 

/(q) = j', and we have 

d^iq) = 2dqqdq-^2q{dqY + 2{dqyq, 

and it is easy to verify by multiplication that we have rigorously 
j'\-xdqy =q* + x{q^dq + qdqq + dqq^)+x^{dqqdq-^q{dqy-^{dqyq) + x^{dqy I 

which is the value given by the application of the above form 
of Taylor's Theorem. 

As we shall not have occasion to employ this theorem, and 
as the demonstrations which have been found are all too la- 
borious for an elementary treatise, we refer the reader to Hamil- 
ton's works, where he will find several of them. 

136. To diflferentiate a function of a function of a quaternion 
we proceed as with scalar variables, attending to the peculiarities 
already pointed out. 

137. A case of considerable importance in geometrical appK- 
cations of quaternions is the differentiation of a scalar Amotion 
of p, the vector of any point in space. 

Let F{p) = (7, 

where jP is a scalar function and C an arbitrary constant, be 
the equation of a series of surfaces. Its differential, 



ia, of course^ a scalar ftinction : and^ being homogeneous and 
linear in dp, § 130, may be thus written, | 

Svdp = 0, I 

where v is a vector, in general a function of p. 

This vector, p, is easily seen to have the direction of the 
normal to the given surface at the extremity of p ; being, in ' 
fact, perpendicular to every, tangent line dp, §§ 36, 98. Its | 
length, when J' is a surface of the second degree, is the re- I 
ciprocal of the distance of the tangent-plane from the origin. | 
And we will show, later, that if 

p = ix-^jy+iz, I 


1. Showtliat 

(a.) d.8Uq = S.UqV^- = "^ i^ ^'^*- 


(e.) d\Tq = {8\dqq-^--8.{dqqr'y}Tq=i -TqV 

2. If Fp = :2.8apSpp + lffp^ 

give dFp = Svdp, 
show that i; = 2 r.ap)3 + (^ + 2 Safi)p. 



138. ^J^TE have seen that the diflterentiation of any fdnc- 
▼ ▼ tion whatever of a quaternion, q, leads to an 
equation of the form 

where/^is linear and homogeneous in dq. To complete the pro- 
cess of differentiation, we must have the means of solving this 
equation so as to be able to exhibit directly the value of dq. 

This general question is not of so much practical importance 
as the particular case in which j^ is a vector ; and, besides, as 
we proceed to show, the solution of the general question may 
easily be made to depend upon that of the particular case ; so 
that we shall commence with the latter. 

The most general expression for the function/ is easily seen 
to be ^ ^^(y^ dj) = 2 Kadqb + S.cdq, 

where a, 6, and e may be any quaternion ftmctions of q what- 
ever. Every possible term of a linear and homogeneous fimc- 
tion is reducible to this form, as the reader may easily see by 
writing down all the forms he can devise. 

Taking the scalars of both sides, we have 

Sdr = 8.cdq = SdqSc +8.VdqVc. 
But we have also, by taking the vector parts, 

Vdr ^'S.V.adqb = Sdq.'lVab-Y'2.V.a{Vdq)b. 

98 QUATERNIONS. [chap. V. 

Eliminating Sdq between the equations for Sdr and Vchr it is 
obvious that a linear and vector expression in Vdq will remain. 
Such an expression, so far as it contains Vdq^ may always be 
reduced to the form of a sum of terms of the type a5./3 Vdq^ by 
the help of formulae like those in §§ 90, 91. Solving this, we 
have Vdq, and Sdq is then found from the preceding equation. 

139. The problem may now be stated thus. 
Find the value of p from the equation 

aSpp + a^Sfi^p-^... = l.aSpp = y, 

where o, /3, Oj, i3i, ...y are given vectors. [It will be shown 
later that the most general form requires but three terms^ 
i. e. six vector constants a, /3, Oi, /3i, a„ /S, in all.] 
If we write, with Hamilton, 

it>p = l.aSpp, 
the given equation may be written 

or p = il>-'y, 

and the object of our investigation is to find the value of the 
inverse function <^-*. 

140. We have seen that any vector whatever may be ex- 
pressed in terms of any three non-coplanar vectors. Hence, 
we should expect a priori that a vector such as <t>^p, or <^'p, for 
instance, should be capable of expression in terms of p, <^p, and 
<^*/). [This is, of course, on the supposition that p, <^p, and 0*p 
are not generally coplanar. But it may easily be seen to extend 
to this case also. For if these vectors be generally coplanar, 
so are <^p, <J)'p, and <p^p, since they may be written <r, <^(r, and 
<f>'(r. And thus, of course, <^'p can be expressed as above. If 
in a particular case, we should have, for some definite vector p, 
<^p = gp where ^ is a scalar, we shall obviously have <^*p = g^p 
and 0'p = g^py so that the equation will still subsist. And a 


similar explanation holds for the particular case when (for some 
defmite value of p) p, <^p, and ^'p are coplanar. For then we 
have an eqiiation of the form 

which gives ^«p = Ail>p + B<p^p 

= ABp'{-{A-\-B^)4>p. 
So that </>'p is in the same plane.] 
If, then, we write 

— <>V = ^P+y#+'8^*V> (1) 

it is evident that x^y^z are quantities independent of the vector 
p, and we can determine them at once by processes such as those 
in §§ 91, 92. 

If any three vectors, as i,y, i, be substituted for p, they will 

in general enable us to assign the values of the three coefficients 

on the right side of the equation, and the solution is complete. 

For by putting <^"*p for p and transposing, the equation becomes 

— a?</)-V = yp + z<l>p'^<l>*p; 

that is, the unknown inverse function is expressed in terms of 
direct operations. If x vanish, while y remains finite, we sub- 
stitute <f>-^p for p, and have 

-y<t>''p:=zp + <t>p, 

and if x and y both vanish 

z<l>-^p = p. 

14L To illustrate this process by a simple example we shall 
take the very important case in which <^ belongs to a central 
surface of the second order ; suppose an ellipsoid ; in which case 
it will be shown (in Chap.VIII.) that we may write 

<^p = — fit* iSip — b^j8;p — c* iSip. 
Here we have 

(f}i = aH, <f>H = a*i, <I>H = aH, 
4>; = b'j, 4>^j = b'j, 4>'j=^i'jy 

o 2 


Hence, putting separately i^j, k for p in the equation (1) of 

last section^ we have 

— «• = x+ya* + i»% 

— <?• s: x+yc*+zc*. 
Hence a% J% c* are the roots of the cubic 

which involves the conditions 

2:=— («• + *•+<?•), 

Thus, with the above value of <!>, we have 

^»p = a»J»c«p— (a•J» + iV+c»a•)0p^-(a» + i* + (?•)<>V• 
142• Putting ^-*<r in place of p (which is any vector what- 
ever) and changing the order of the terms, we have the desired 
inversion of the function <f> in the form 

or the inverse function is expressed in terms of the direct 
function. For this particular case the solution we have given 
is complete, and satisfactory ; and it has the advantage of pre- 
paring the reader to expect a similar form of solution in more 
complex cases. 

143. It may also be useful as a preparation for what follows, 
if we put the equation of § 140 in the form 

= {(</>-«')(*-J')(*-^')}p .' (2) 

This last transformation is permitted because <f> is commutative 
with scalars like a% i. e. 0(«'p) == «'0p- 


Here we remark that (by § 140) the equation 
V.ixl>p = 0, or </)p = ffpf 
where ff is some undetermined scalar^ is satisfied^ not merely by 
every vector of null-length, but by the definite system of three 
rectangular vectors Ai, Bj, Ck whatever be their tensors, the 
corresponding particular values of ^ being a*, }», c*. 

144. We now give Hamilton's admirable investigation. 
The most general form of a Unaar and vector fimction of a 

vector may of course be written as 

^p = 2 y^qpr, 
where q and r ai-e any constant quaternions. 

Hence, operating by S.v where a is any other vector, 

S<T<l>p = 2S.(^Spit/<r, (3) 

if we agree to write 

4>^ar = iV.ra-q. 
The functions <f> and (j/ are thus eofyugate to one another, and 
on this property the whole investigation depends. 

145. Let X, /A be any two vectors, such that 

Operating by 8X and 8,ii we have 

5X0P = 0, fiJLu^p = 0. 
But, introducing the conjugate function 0^, these become 

8p4f\ = 0, iSp^V = 0, 
and give p in the form 

mp = F</)'A<^V> 
where ^ is a scalar which, as we shall presently see, is inde- 
pendent of \, fi, and p. 

But our original assumption gives 

P = <>-*FVr 

hence we have 

OT<^-' FA/i = F<>'X0V, (4) 

and the problem of inverting ^ is solved. 



[chap. V. 


146* It remains to find the value of tlie constant m, and 
to express the vector 

as a function of FA/a. 

Operate on (4) by 8.(f/vf where v is any vector not cophtnar 
with X and fj,, and we get 

mS.<l/pil>-^ FV = fnJS.v<t><ir^VKii (by (3) of § 144) 

= mS.\iiv= S.<l/\it/iMl/v, or 

"* "" 8.\,xp 

[That this quantity is independent of the particular vectors 
\, IX, via evident from the fact that if 

\'=jpX+j/i+ry, M'=i'A+S'iM+^i»'* and i/=p^k + q^fi+r^v 
be any other three vectors (which is possible since X, ^i, v are 
not coplanar)^ we have 

<t>h^ = p4>k + j^V + r^fv, &c., &c. ; 
from which we deduce 



so that the numerator and denominator of the fraction which 
expresses m are altered in the same ratio. Each of these Quan- 
tities is in fact an Invariant, and the numerical multiplier is the 
same for both when we pass from any one set of three vectors 
to another.] 

147. Let us now change <^ to <>+^j where ^ is any scalar. 
It is evident that <^' becomes <^'4-^, and our equation (4) be* 






















= F^'A^V -^g r{4>'\fi + X0V) -e^' FA/a, 
= (»^<J)-*-|-^X+^'')F^M suppose. 
In the above equation 

is what m becomes when </> is changed into </>+^; ^i and ^^ 
being two new scalar constants whose values are 

mi = ^ 9 


Substituting for Ug, and equating the coefficients of the various 
powers of ^ after operating on both sides by (f>-\-g, we have two 
identities and the following two equations^ 

[The first determines x> ai^d shows that we were justified in 
treating F(<^'A/i+A<^V) ^ ^ linear and vector function of FA/a, 
The result might have been also obtained thus, 

SAxVXfx = 5.A^'A/A = -&Afi0'A = -/S.A^FA^, 

8.IXxVXtA = 8.lik<t>'fA=z^S.IX(j>V\lM, 

= maS\fiv—S.\fjxl/v 
= 5.v(w,FA/A-<>FX/ui); 
and all three are satisfied by 

X = «^,-0.] 

148. Eliminating x &om these equations we find 

Ml = (^(^a — <^) + W0~S 
or «»<^- ' = l»i — «»j<;f) -f <J)% 


which contains the complete solution of linear and vector equa- 

149. More to satisfy the student of the validity of the above 
investigation^ about whose logic he may at first feel some diffi- 
culties^ than to obtain easy solutions^ we take a few very simple 
examples to begin with: and we append for comparison easy 
solutions obtained by methods specially adapted to each case. 

150. Example!. 

Let <^p =: V.apfi = y. 
Then ^'p = V.fipa = ^p. 

Hence m = -^^ S{V.aXp F.ofi/S V.avp). 

Now \y ii, V are any three non-coplanar vectors ; and we may 
therefore put for them o, P,yiftke latter be non-coplanar. 
With this proviso 


m^ = -=-— 8{a^p.aPKy+a.apKr.ayP + a^p.p.V.ayfi) 

a^P*SaP4-'yz^^a*P^8afi.p^ ^a*fi^y+SapV.ayP+ r.a{V.ayp)fij | 
which is one form of solution. 

By expanding the vectors of products we may easily reduce i 
it to the form 

a^p^8afi.p = - a«^ V + a)3«/Say .+ fia^S^y, 

or p= —Saf^^- 


15L To verify tbis solution^ we have 

which is the given equation. 

152. An easier mode of arriving at the same solution^ in this 
simple case^ is as follows : — 

Operating by 8.a and on the given equation 

V.apfi = y, 

we obtain a^Sfip = Say, 


and therefore aSpp = a-^Say, 

pSap = P'^Spy. 

But the given equation may be written 

aSPp—pSafi-^-pSap = y. 

Substituting and transposing we get 

pSafi = o-»Say+/3-*/8)3y-y, 

which agrees with the result of § 150. 

153. K a, /3, y be coplanar, the above mode of solution is 
applicable^ but the result may be deduced much more simply. 

For (§101) 8.aPy=0, and the equation then gives 8.afip=^0, 
so that p is also coplanar with a, fi, y. 
Hence the equation may be written 

apP = y, 
and at once 

and this^ being a vector^ may be written 

= a-'Sp-'y'\-p-'8a-'y-y8a-'p-\ 


This fonniila is equinaleni to that just given^ bat not eqxuil 
to it term by term. [The student will find it a good exercise 
to prove direeUy that^ if Oj ^> y are coplanar, we have 

154. Example 11. 

Let <^p = V.afip = y. 

Suppose a, p, y not to be coplanar^ and employ them 2i&\,ii,v 
to calculate the coefficients in the equation for ^~'. We have 

8.ir4>p = S.fTcfip = B.pV.traP = S.pif/a. 

Hence tj/p = V.pap = V.fiap. 

We have now 

«»i = •^^iS(a.i3o)8.F./3oy+/3a».0.F./3ay+i8o»./3a/3.y) 

= 2(iSai3)« + a»/3% 

^a = ^;^^(M.^.i3ay + a.)3ai3.y+/3o\^,y) 

= 3fifa)3. 
a»/3«/So/3.^-^y = a^fi^Safi.p 

= (2(&/3)« + a«/3«)y-3&^F.o/3y4- V.apV.aPy, 
wbich^ by expanding the vectors of products^ takes easily the 
simpler form 

a^fi'Sap.p = a*^*y-a)3«flfay + 2/3/S(i^fifay-i3a«/Sr^y. 

155. To verify this, operate by V.a^ on both sides> and we 



+ 2afi^S€Lfi8ay-aa*fi*Spy 

or V.afip = y, 

156. To solve the same equation without employing the 
general method^ we may proceed as follows : — 

y = F.o/3p = p8ap + r. F(a/3)p. 

Operating by 8. Vafi we have 

8.afiy = S.afipSafi. 

Divide this by fiia/S, and add it to the given equation. We 
thus obtain 

Hence p = ^.a-(y+^), 

a form of solution somewhat simpler than that before obtained. 

To show that they agree^ however, let us multiply by a*/3'/Sa)3, 

and we get 

a^fi*8afi.p = fiay8aP'\'pa8.afiy. 

In this form we see at once that the right-hand side is a vector, 
since its scalar is evidently zero (§ 89). Hence we may write 

a^fi*8afi.p = F.^ayfifa/3— Vafi8.afiy. 
But by (3) of § 91, 

-y/8^.o/3Fa^ + aS'./3(Fa^)y + ^fl^.F(a^)ay+ VafiS.a^y = 0. 
Add this to the right-hand side, and we have 
a'P^Sap.p z=,y{{8afiyS.apraP)-a{{rafi)y) 

+ fi{Sap8ay + 8.V{aP)ay), 
P % 

108 QUATBRNION& [chap. V. 

But {8apyS.apraP = (iSaj8)»-(ra/3)» = a»/3% 

Sap8pyS.p{rap)y = SapSpySpaSpyk-p^Say = ^^Say 

Sap8ay^8.F{ap)ay = 8ap8ay+8ap8ay'~a*8py 

= 2iai0iSoy-.a«i8)3y; 

and the substitation of these values renders our equation identical 
with that of § 154. 

[If a, p,y he coplanar^ the simplified forms of the expression 
for p lead to the equation 

8afi.^^a''y = y— o-»fifoy-f 2^/So-'/3-»iSay— /3-*/8)3y, 

which, as before, we leave as an exercise to the student.] 

157. Sample III. The solution of the equation 

leads to the vanishing of some of the quantities m. Before, 
however, treating it by the general method, we shall deduce its 
solution from that of rr a 

already given. Our reason for so doing is that we thus have 
an opportunity of showing the nature of some of the cases in 
which one or more of m, m^, i», vanish ; and also of introducing 
an example of the use of vanishing fractions in quaternions. 
Far simpler solutions will be given in the following sections. 

The solution of the last- written equation is, § 154, 

a^$^8ap.p = a^p^y-aP^8ay^Pa^8py+2p8afi8ay. 

If we now put 

0/3 = e + € 

where e is a scalar, the solution of the first-written equation 
will evidently be derived fix)m that of the second by making e 
gradually tend to zero. 

We have, for this purpose, the following necessary trans- 
formations : — 

a'^jS* = alBK.a^ = (^4.e)(^— c) = ^» — e% 


afi^Say-^pa^Spy = afi.pSay -^ pa.a8py, 

= {e + €)fi8ay'^{e^€)a8fiy, 

= ^(/3/Soy + aSpy) -f € Fyc. 
Hence the solution becomes 

(e«_€>)^p = (<?«— €«)y— ^(/354iy+oS)8y)-cFy€ + 2tf/3/8oy, 
= (<?«-€»)y+<?F.yFa)8-€Fy€, 
= («•— c')y+^Fy€ + y€"— c/Syc, 
= ^*yH-^Fyc— €/8y€. 
Dividing by e, and then putting d = 0, we have 

Now, by the form of the given equation, we see that 

8y€ = 0. 
Hence the limit is indeterminate, and we may put for it x, 
where x is any scalar. Our solution is, therefore, 

p = -F^+^€-»; 

or, as it may be written, since 8y€ = 0, 

p = €-»(y+«). 
The verification is obvious — for we have 

€p szy-k-X. 

158. This suggests a very simple mode of solution. For 

we see that 

F€p = F€(p-a?€-0 = y, 

a constant vector whatever x may be. But the vector sign may 
now be removed as unnecessary, so that we have 

€{p-X€-^) = y, 

or p = €-Hy + ^)j 

if, and only if, p satisfies the equation 


110 QUATERNI0K8, [cHAP. V. 

159* To apply the general method, we may take c, y and cy 
(which is a vector) for A, \Ly v. 

We find ^> =s Vp€. 
Hence m =s 0, 

«ii = - — S.(€,ey.€»y) = -€% 

»i, = 0. 

Hence — c*^ + ^» = 0, 


*"* = ^*+*"*^- 

That is, P = ~i^ ^«y+^€> 

= 6"*y+jr€, as before. 

Our warrant for putting X€ as the equivalent ottft'-'O is this : — 
The equation ^.^^^ 

may be written 

F.fFco- = = <rc* — c&<r. 

Hence, unless o- = 0, we have o* || € = X€. 

160. Example IV, As a final example let us take the most 
general form of <^, which, as will be soon proved, may be ex- 
pressed as follows : — 

i>p = aSPp-^-a^Sp^p + a^Sp^p = y. 
Here fp'p = pSap + PiSa^p + P^Sa^p, 

and, consequently, taking a, a^, a„ which are in this case non- 
coplanar vectors, for A, ijl, v, we have 

m = ^ — S,(fi8ajaL+p^Sa^a'{-p^8a^a) {pBaa^^-P^Sa^a^ + ...){fi8ajii^ + . 



Saa Saia Sa^a 
Scuii iSixiai Sa^ai 
Saa^ /SaiOs Sa^a^ 

{ASaa + AiSaia+AtSaiO), 


where A = fi^aiai Sa^a^ — Sa^ax So,^'^^ 

= — £•. Fctjaa Fiiiaa 
Ax =)Sa,ai iSaaj — iSoaiiSxaaj 

= — /S. FajoFaiaa 
^a = Saai jSaitta — SaiaiSaa^ 
= — iSl Faai FaiOa. 
Hence the value of the determinant is 
— {SauiS. FaiOa Fajaa + Sa^aS. Foaa Foia, + Sd^aS. Faa, Faia.) 

= -iS:a(FoiaaS:aa,aa)*{by § 92 (4)} = -.(fif.aa.aa)^ 

The interpretation of this result in spherical trigonometry is 

very interesting. 

By it we see that 

m =^8.aaxa^S.pfixfi^. 

»i= ^ SlaipSaax +/3i&iai -^ p^Sa,ax){pS^a, + PxSaxa,+p,Sa,a,) + &c.] 


= -^ (&appx{SaaiSaxa^''SaxaxSaa^)+ ) 


^ (&a/3^i)S.aF.a,Faaai + ) 


= - -jT^^ [S,a( VfiM^ Faai Fata, + Vp,pS. Va^aVa^a^ + Vfixfi.S. Va^a^ Fa,aa) 


+{VpfixS.VaaxVa,a+ ) 

+ S.a,{Vfifi,8.VaaxVaax + )]; 

or, taking the terms by columns instead of by rows, 

= - -^ [S. VfiPx {aS. Vaux Va^ai + a^S. Faa^ FaaO + a^S. Faa, Faa^ 

+ + ], 



or, grouping as before, 

= - -^ — S [/3(rao,flifta, + Va^aSaa^ + ra,a,Soa)+ ...], 

* ■8[fi{a8.aa,a,) + ] (§92(4)), 

And the solution is, therefore, 
-^S.aaia^S.pPiPt.fji-^y = ^8.cuiiai8.pfiiPt'p 

[It will be excellent practice for the student to work out in 
detail the blank portions of the above investigation, and also to 
prove directly that the value of p we have just found satisfies 
the given equation.] 

16L But it is not necessary to go through such a long 
process to get the solution — ^though it will be advantageous to 
the student to read it carefully — ^for if we operate on the equa- , 
tion by 8.aia2, iS^.a^a, and 8,aa^ we get | 

S.aia^aSpp = S.a^aiy, \ 

S.a^aaiSpiP = S^^ay, \ 

S.aaia^SfitP := 8,aaiy. 
From these, by § 92 (4), we have at once 

The student will find it a useful exercise to prove that this is 
equivalent to the solution in § 160. 

To verify the present solution we have 
8.aaia,8.pp^p, {aSfip + a,8p,p + a,8M 

= aS.pp^pAaia^y + a^S.^^fi^pS^^ay + a^S.^^pp^SM^y 
= 8.pp,p,{y8.aa,a,), by § 91 (3). 


162. It is evident^ from these examples^ that for special cased 
we can usually find modes of solution of the linear and vector 
equation which are simpler in application than the general 
process of § 148. The real value of that process however con- 
sists partly in its enabling us to express inverse functions of <l>, 
such as (0+^)""* for instance, in terms of direct operations, a 
property which will be of great use to us later; partly in its 
leading us to the fundamental cubic 

<^' — »^a^* + «»i0 — «a = 0, 

which is an immediate deduction from the equation of § 148, 
and whose interpretation is of the utmost importance with 
reference to the axes of surfaces of the second order, principal 
axes of inertia, the analysis of strains in a distorted solid, and 
various similar enquiries. 

163. When the function (f> is its own conjugate, that is, when 
for all values of p and a, the vectors for which 

form in general a real and definite rectangular system. This, 
of course, may in particular cases degrade into one definite 
vector, and any pair of others perpendicular to it j and cases 
may occur in which the equation is satisfied for every vector. 
Suppose the roots of %=0 (§ 147) to be real and different, then 

#1 = giPi ^ 

^Pi = ^aPa " where p^, p,, p, are definite vectors. 

#s = 9zP% - 

Hence gig^Sp^p^ = S.t^p^i^p^ 

= S.p.^^p^, or = S.p^(l>^p„ 
because <^ is its own conjugate. 
But <p^p, = glp^, 

<^Vi = glpi> 


and therefore 

which^ as ^, and ^, are bj hypothesis different, requires 

Spipi = 0. 

Similarly Sp^p^ = 0, Sp^i = 0. 

If two roots be equal, as ^,, ^,, we still have, by the above 
proof, SpiPi = and Spip^ = 0. But there is nothing farther 
to determine p, and p„ which are therefore any vectors per- 
pendicular to pi. 

If all three roots be equal, every real vector satisfies the equa- 
tion (<^— ^)p = 0. 

164. Next, as to the reality of the three directions in this 

Suppose ^,+*a\/— 1 to be a root, and let p^ + o■,^/— 1 be 
the corresponding value of p, where ^, and i^ are real numbers, 
p, and 0-, real vectors, and a/— 1 the old imaginary of algebra. 

Then *(p, + (r,A/^) = (^,+*,A/3T)(p, + <r,x/^), 
and this divides itself, as in algebra, into the two equations 

Operating on these by S.<r,j 5.p, respectively, and subtracting 
the results, remembering our condition as to the nature of ^ 

5<r,<^p, = 5p,^„ 
we have ^a(<^J+Pi) = 0. 

But, as (Ts and p, are both real vectors, the sum of their 
squares cannot vanish. Hence it vanishes, and with it the 
impossible part of the root. 

165. When ^ is self-conjugate, we have shewn that the 

has three real roots, in general different &om one another. 


H^iee the cubic in tft may be written 

and in this form we can easily see the meaning of the cubic. 
For, let pi, p,, p, be such that 

Then since any vector may be expressed by the equation 
pS.pxpipt = pAp^tP+pAptpip-^PiS-piPip (§ 91), 
we see that when the complex operation, denoted by the left- 
hand member of the above symbolic equation, is performed on p, 
the first of the three factors makes the term in pi vanish, the 
second and third those in p, and p, respectively. In other 
words, by the successive performance upon a vector of the opera- 
tions <^— ^1, <^— ^„ <^— ^8> it is deprived successively of its 
resolved parts in the directions of pi, p,, p, respectively ; and 
is thus necessarily reduced to zero, since pi, pa, p. are (because 
we have supposed ffi, ^„ ^, to be distinct) rectangular vectors. 

166. If we take pi, p„ p, as rectangular unU-Yecbois, we have 

— p = piSpip-^-p^Sp^p-^p^Sp^p, 
whence 4>p = — ^ipj^pip —gtptSp^p SftptSp^p ; 
or, still more simply, putting i,j\ k for p,, p„ p,, we find that 
cmy self-conjugate function may4)e thus expressed 

<lip ^-giiSip—gJSjp-^g^kSkp, 
provided, of course, i,j, k be taken as roots of the equation 

Vp4^p = 0. 

167. A very important transformation of the self-conjugate 
linear and vector function is easily derived from this form. 

We have seen that it involves three scalar constants only, 
viz- gi} g\9 ^8« I^t ^ enquire, then, whether it can be reduced 
to the following form 

# =pP'^qV.{i'\-ek)p{i'-'ek), 
which also involves but three scalar constants j9, q, e. 



Sabstituting for p the equiyalent 

p = ^iSip^jSjp-^kSkpy 
expanding, and equating coefficients of ij j, h in the two ex- 
pressions for ^py we find 

-^i=-/' + ?(2- ! + <?»), 
-y, =-j?-j(l-<r«). 

These gire at once 

—(^2—^3)= ^e*q. 
Hence, as we suppose the transformation to be real, and therefore 
«» to be positive, it is evident that gx^g% and ^,—^8 have the 
same sig^ ; so that we must choose as auxiliary vectors in the 
last term of ^p those two of the rectangular directions i,y, k for 
which the coefficients g have the greatest and least values. 
We have then 

i = -\{gx-g^y 

and jD = i(^i+ii's). 

168. We may, therefore, always determine definitely the 
vectors X, /m, and the scalar jd, inr the equation 


when ^ is self-conjugate, and the corresponding cubic has not 
equal roots, subject to the single restriction that 

is known, but not the separate tensors of X and p.. This result 
is, important in the theory of surfaces of the second order, and 
will be developed in Chapter VII. 

169. Another important transformation of </> when self-con- 
jugate is the following, 

<^p = aa Vap + h^S^py 


where a and b are scalars^ and a and fi are unit-yectors. This^ 
of course^ involves six scalar constants^ and belongs to the 
most general form 

where pi, p^, ps are the rectangular unit-vectors for which p 
and <t>p are parallel. We merely mention this form in passing, 
as it belongs to the /ocal transformation of the equation of 
surfaces of the second order, which will not be farther alluded 
to in this work. It will be a good exercise for the student 
to determine a, j8, a and 8, in terms of ^i, ^„ ^j, and pi, p,, p,. 

170, We cannot afford space for a detailed account of the 
singular properties of these vector functions, and will content 
ourselves with the enunciation and proof of one or two of the 
most important. 

In the equation 

«»0-«rx|A = r</)'x0V (§ 145), 

substitute X for ^^X and n for <f/[x, and we h*ave 

Change <^ to <^+^, and therefore <^' to ^'+y, and m to %, 

a formula which will be found to be of considerable use. 

17L Again, by § 147, 

^8.p{^i,+h)-^p = ^^p^-'p + 5px)»+ V- 

^«.p(<^+^)-V - f &p(* + A)-«p = {g-h){p^- ^^} . 


That is^ the fanctions 

^M*+^)-V> and ^Ap(<^+*)-ip 

are identical, i. e. wien equaled to eoMiawU represent the same 
series of surfaces, not merely when 

but also, whatever be g and i, if they be scalar functions of p 
which satisfy the equation 

This is a generalization, due to Hamiltcm, of a singular result 
obtained by the author *. 

172. The equations 

^•p(*+^)-V = o,-i ^jj 

are equivalent to 

m8p<l>-*p+gSpxp+g*p* = 0^ 

mSpip-^p-i-iSpxp-^i^p' = 0. 
Hence «(1— ar)i8jp<^-*p+(^— AF)iSpXP+W*""**^)/»* = ^> 
whatever scalar be represented by x. 

That is, the two equations (1) represent the same surface if 
this identity be satisfied. As particular cases let 

(1) 0? = 1, in which case 

(2) y— ^ar = 0, in which case 

or mSp^^i^r^p'^gh = 0. 

* Note on the Cartesian equation of the Wave-Surface. Quarterly Mathem, 
Journal, Oct. 1859. 


(3) ^=j^> giving 

or m{A +ff)8p<f}-^p -k-ghSpyj) = 0. 
173. In various investigations we meet with the quaternion 

where a, fi, y are three unit- vectors at right angles to each 
other. It admits of being put in a very simple form^ which 
is occasionally of considerable importance. 

We have^ obviously, by the properties of a rectangular unit- 
system J _ py^^ ^ y^^p ^ ^p^^ 

As we have also 

&ai3y = -l (§71(13)), 

a glance at the formulae of § 147 shows that 

at least if <^ be self-conjugate. If it be not, then (as will be 
shown in § 174) 

4>p = 4/p+ F€p, 

and the new term disappears in 8q. 
We have also, by § 90 (2), 

Fq = a{8p4>y'^SY<lip)+p{SY4>a-'8cufyy)-^y{Sa<liP'-Sfi<tia) 

= a8.p€y+pS.y€a-\-y8M€p 

= — (a&ic+i3fil3€ + ySyc) = €. 

Many similar singular properties of ^ in connection with a 
rectangular system might easily be given ; for instance, 

r{aF<l>fi4>y^pr4yy<lia+yF4>a4>fi) = ij>€; 

which the reader may easily verify by a process similar to that 
•just given, or (more directly) by the help of § 145 (4). A few 


others will be found among the Examples appended to this 

174. To conclude^ we may remark that as in many of the 
immediately preceding investigations we have supposed <^ to be 
self-conjugate^ a very simple step enables us to pass from this 
to the non-conjugate form. 

For, if ^' be conjugate to if>, we have 
8p(l/a = 8a'<f>pf 
and also SpifHr = Satj/p. 

Adding, we have 

Sp(4> + 4>>-Sa{4>+<t/)p; 
80 that the function (0-1-^0 ^ self-conjugate. 

Again, Sp4>p = Spff/p, 

which gives 'Sp{<t>^40p = ^• 

Hence {<t>^4>')p = ^^Pt 

where c is some real vector, and therefore 

Thus every noU'-conjugate linear and vector function differs from 
a conjugate function solely by a term of the form 


The geometric signification of this will be found in the Chapter 
on Kinematics. 

175. We have shown, at some length, how a linear and 
vector equation containing an unktiown vector is to be solved 
in the most general case; and this, by § 138, shows how to 
find an unknown quaternion from any sufficiently general linear 
equation containing it. That such an equation may be suffi- 
ciently general it must have both scalar and vector parts : the* 


first gives one^ and the second three j scalar equations ; and these 
are required to determine completely the four scalar elements 
of the unknown quaternion. 

17a Thus Tq^a 

being* but one scalai^ equation^ gives 

where r is any quaternion whatever. 

Similarly Sq = a 

gives q = a + 0, 

where is any vector whatever. In each of these eases^ only 
one scalar condition being given, the solution contains three 
scalar indeterminates. 

177. Again, the reader may easily prove that 

where a is a given vector, gives 

Hence, assuming 

Saq ^y, 

We have aq ^y-j-xa+fi, 

or J = ar+jfa~* + o~*)8. 

Here, the given equation being equivalent to two scalar con- 
ditions, the solution contains two scalar indeterminates. 

178. Next take the equation 

Operating by Ao-*, we get 

8q = 8a-' p, 
so that the given equation becomes 

F'a(&-^i3+rj) = /3, 
or . FaFq = p-^a8a''P = af^a'^fi. 



Prom this, by § 168, we see that 

whence q — Sa-^fi-^ a-' {x + aFit-'^) 

and, the given equation being equivalent to three scalar con- 
ditions, but one undetermined scalar remains in the value of q. 

This solution might have been obtained at once, since our 
equation gives merely the vector of the quaternion aq, and leaves 
its scalar undetermined. 

Hence, taking x for the scalar, we have 

aq = Saq-i- Vaq 

= x+fi, 

VI9. Finally, of course, from 


which is equivalent to four scalar equations, we obtain a definite 
value of the unknown quaternion in the form 

q = a-'p. 

180. Before taking leave of linear equations, we may mention 
that Hamilton has shown how to solve any linear equation 
containing an unknown quaternion, by a process analogous to 
that which he employed to determine an unknown vector from 
a linear and vector equation ; and to which a large part of this 
Chapter has been devoted. Besides the increased complexity, 
the peculiar feature disclosed by this beautiful discovery is that 
the symbolic equation for a linear quaternion function, cor- 
responding to the cubic in c^ of § 162, is a biquadratic, so that 
the inverse function is given in terms of the first, second, and 
third powers of the direct function. In an elementary work 
like the present the discussion of such a question would be 
out of place : although it is not very difficult to derive the 
more general result by an application of processes already ex- 


plained* The reader is therefore referred to the EkmenU of 
-QuatemionSy p. 491. 

181, The solution of the following frequently-occurring par- 
ticular form of linear quaternion equation 

aq-^-qh = <?, 
where a, b, and c are any given quaternions^ has been effected 
by Hamilton by an ingenious process, which was applied in 
§ 133 (5) to a simple case. 

Multiply the whole by Ka, and into b, and we have 

T^a.q^- Ka.qb = Ka,c, 
and aqb + qb^ = cb. 
Adding, we have 

q{T^a + 6* -f 2 8a,b) = Ka.c + cb, 
from which q is at onde found. 
To this form any equation such as 
a'qV'{'c'qcF= e' 
can of course be reduced, by multiplication by (dY^ and into 

182. As another example, let us find the differential of the 
cube root of a quaternion. If 

J* = r 
we have q^dq + qdq,q + dq.cj^ = dr. 

Multiply by q, and into q-^, simultaneously, and we obtain 

q^dq-q'^-^q^dq + qdq.q = qdr.q~^. 
Subtracting this from the preceding equation we have 
dq»q^—q^dq,q~^ = dr—qdr,q-^, 

from which dq, or d(r^), can be found by the process of last 

The method here employed can be easily applied to find the 
differential of any root of a quaternion. 

R 7, 


183. To show some of the oharacteristie peouliarities in the 
solution even of quaternion equations of the first d^;Tee when 
they are not sufficiently general^ let us take the very simple one 

and give every step of the solution, as practice in transforma- 
Apply Hamilton's process (§181), and we get 

T^a,q = Ka.qb, 
qb* = aqb. 
These give q{T'a+b*-^2b8a) = 0, 

so that the equation gives no real finite value for q unless 
T*a-\-b^^2bSa = 0, 
or b^^Sa^pTVa 
where p is some unit- vector. 

By a similar process we may evidently show that 
a=zSb + aTrb, 
a being another unit- vector. 
But, by the given equation, 

Ta = Tb, 
or 8^a+T^Fa=:8^b + T^Fb; 
from which, and the above values of a and i, we see that 

Sa 8b 

Wa'^Wb^''' '^PP^'"- 

If, then, we separate q into its scalar and vector parts, thus 

the given equation becomes 

(a + a)(r + p) = (/• + p)(a + i9) .'.J. (I) 

Multiplying out we have 

r(o-^) = pfl-ap. 


which gives 8{v^'^fi)p = 0, 

and therefore p^s Fy{a''P)y 

where y is an undetermined vector. 
We have now 

r{a-p) = p/3-ap 

= -(a-^)^(a-h^)y. 
Having thus determined r, we have 

2?=-(a+/3)y-y(a-h/3) + y(a-^)-(a-0)y 
= — 2ay — 2y^. 
Here, of course, we may change the sign of y, and write the 
solution of aq^qb 

in the form q = ay+yp, 

where y is any vector, and 

a = UVa, p = ?7ri. 
To verify this solution, we see by (1) that we require only 
to show that aq = jj3. 

But their common value is evidently 
— y+ay^. 

It will be excellent practice for the student to represent the 
terms of this equation by versor-arca, as in § 54, and to deduce 
the above solution from the diagram directly. He will find 
that the solution may thus be obtained almost intuitively. 

184. No general method of solving quaternion equations of 
the second or higher degrees has yet been found; in fact, as 
will be shown immediately, even those of the second degree 
involve (in their most general form) algebraic equations of the 


sixteenth Aegre^. Hence, in the kvf remaining section? of tim 
Chapter we shall confine ourselves to one or two of ilie' simple 
forms for the treatment of which a definite process has been 
devised. But first, let us consider how many roots an equation 
of the second degree in an unknown quaternion must generally 

If we substitute for the quaternion the expression 
w^-ix-^jy+kz (§ 80), 

and treat the quaternion constants in the same way, we shall 
have (§ 80) four equations, generally of the second degree^ to 
determine w, x, y, z. The number of roots will therefore be 
2* or 16. And similar reasoning shows us that a quaternion 
equation of the «»th degree has m* roots. It is easy to see, 
however, from some of the simple examples given above (§§ 175 
^178, &c.) that, unless the given equation is equivalent to four 
scalar equations, the roots will contain one or more indeterminate 

185. Hamilton has effected in a simple way the solution of 

the quadratic 

j« = qa-^rby 

or the following, which is virtually the same (as we see by taking 
the conjugate of each side). 

He puts S' = i(^+^+p) 

where «e; is a scalar, and p a vector. 
Substituting this value, we get 

a^ -{■{w-^pY-^'^wa + ap-^'pa = 2{a^-\-wa'\-pa)-\-^b, 
or {w-{'py-{'ap'-pa = a'-h4i. 
If we put Va = a, %»-f 4i) = c, V{a^ + ^b) = 2y, this be- 
^^"'^^ («;+/>)'-h2rap = ^+2y; 


wfaich^ by equating separately the scalar and vector parts^ may 
be broken up into the two equations 

The latter of these can be solved for p by the process of § 156, 
or more simply by operating at once by 8,a which gives the 
value of 8{W'^a)p. If we substitute the resulting value of 
p in the former we obtain, as the reader may easily prove, the 

The solution of this scalar cubic gives six values of w, for each 
of which we find a value of p, and thence a value of q, 

Hamilton shows {Lectures , p. 633) that only two of these 
values are real quaternions, the remaining four being biqua- 
temions, and the other ten roots of the given equation being 

Hamilton farther remarks that the above process leads, as 
the reader may easily see, to the solution of the two simul- 
taneous equations 

and he connects it also with the evaluation of certain continued 
fractions with quaternion constituents. 

186. The equation 

g^ = aq-k-qby 

though apparently of the second degree, is easily reduced to 
the first degree by multiplying by, and into, qr^y when it be- 
comes l=^ia + J^-i, 
and may be treated by the process of § 181, 

187. The equation 



where a and b are given quatemions and A^ a scalarj gives 

^^7 qaqb^aqbq (1) 

This may be written 

Iq^aqb) = {aqb)q; 

&>^d> ^7 § ^^9 it is evident that the planes of q and aqb must 
coincide. A little fart^ier consideration will show that in general 
we must have the planes of a, by and q coincident. The solution 
of such equations th|ks becomes very easy, for the commutative 
law of multiplication is not violated (§ 54). 


^ !. Solve the following equations : — 
{a.) = r.ayp. , 
(«.) apfip = papp. 
(<?.) ap+pP=: y. 
{d.) S.aPp+fiSap'-firfip = y. 
{€.) p + apfi^afi. 

Does the lasi of these impose any restriction on the generality 
of a and /3? 

/ 2. Suppose p^i^+jy-k-kzy 

and 4>p = aiSip + bjSjp -f ckSkp ; 
put into Cartesian coordinates the following equations :-— 
(a.) T<t>p = 1. 
(i.) 8p<l>^p = ^l. 

{d.) Tp = T.<f>Up. 


3. If kyfjL, V he any three non-coplanar vectors, and 

show that q is necessarily divisible by SX^uf. 
Also show that the quotient is 

where Fep is the non-commutative part of ^p. 

Hamilton, ElemenU, p. 442. 

4. Solve the simultaneous equations : — 

^ ^ 4Sp<^p = 0. J 
Sap = 0,' 

^«p =0,-. 
^ ' 8.aipKp- O.i 

5. If # = ^fi8ap+ Vrp, 
where r is a given quaternion, show that 


Lectures, p. 561, 

6. If jjoj'] denote pq'-qp, 

(pq) ... S.plqr-], 

Ipqr] ... (i>sr)+[rj]^ + MSj+M^, 
and (pg^^) "' 8. piers'], 

show that the following relations exist among any five qua- 

= JO (qrst) + q {rsfp) + r{8tpq) + s{tpqr) + t{pqf8), 

and jCiw^O = [^*CI^2"" i^iplSrq + [^]%— [jor«]%. 

Elements, p. 492. 



7. Show that if ^^ ^^ be any linear and vector fdnctions^ and 
a, P, y rectangular unit- vectors, the vector 

is an invariant. 

If ^p = 2i,^p, 

and # = S«?ifiiriP, 

show that this invariant may be expressed as 

-sr#f or srt^iKi. 

Show abo that ^^p— ^^/> = VOp. 

8. Show that if 

^p = a8ap'\-pSPp-\-ySypy 
where a,P,y sire any three vectors, then 

^8Kcfiy4-'p = a.Sa.p^p^Sp^p'^ 
where Oi = F/Sy, &c. 

9. Show that any self-conjugate linear and vector function 
may in general be expressed in terms of two given ones, the 
expression involving terms of the second order. 

Show also that we may wri£e 

where a, b, c, x, y, z are scalars, and «r and cu the two given 

10. Solve the equations : — 

[a.) J* = Sji+ioy. 
{L) j« = 2^ + i. - 
{c) qaq ^bq+c. 
(d,) aq z:z qr = rb. 



188. TTAVING, in the five preceding Chapters, given a 
•*--■• brief exposition of the theory and properties of 

quaternions, we intend to devote the rest of the work to ex- 
«amples of their practical application, commencing, of coarse, 
with the simplest curve and surface, the straight line and the 
plane. In this and the remaining Chapters of the work a few 
of the earlier examples will be wrought out in their fiillest 
detail, with a reference to the first five whenever a transforma- 
tion occurs ; but, as each Chapter proceeds, superfluous steps 
will be gradually omitted, until in the- later examples the fiill 
value of the quaternion processes is exhibited. 

189. Before proceeding to the proper business of the Chapter 
we make a digression in order to give a few instances of ap- 
plications to ordinary plane geometry. These the student may 
midtiply indefinitely with great ease. 

(a,) Emlid, I. 5. Let a and ^ be the vector sides of an 
isosceles triangle; j9— a is the base, and 

Ta = T/3. 

The proposition will evidently be proved if we show that 

a{a^P)-^ = Kpifi^a)'^ (§ 52). 
S 2 


This gives a(a-^)-* = 03-a)-»ft 

or (/3-.a)a = i9(a-i9), 
or — o*=— /3*. 

(b.) Huelid, I. 32. Let ABC he the triangle, and let 

AB ^ 

where y is a unit-vector perpendicular to the plane of the 
triangle. If I = \, the angle CAB is a right angle (§ 74). 

Hence ^ = /-(§ 74). Let-B = «»^, C'rr:*-. We have 

2 A 2 

[7CB = y«[7CZ, 
UBA = y~?^5C: 
Hence UBA^ y^.y\'/UAB, 

or — 1 = y'+«+*. 

That is ;+«»+^= 2, 

or A-^B+C=Tr. 

This is, properly speakings Legendre's proof; and might have 
been given in a far shorter form than that above. In fact we 
have for any three vectors, 

U.^ll^l (§60), 
which contains Euclid's proposition as a mere particular case. 

{c.) Euclid, I. 35. Let )3 be the common vector-base of 
the parallelograms, a the conterminous vector-side 
of any one of them. For any other the vector-side 
is a+;r)3 (§ 28)^ and the proposition appears as 

Trp{a-\-xfi) = TFpa (§§ 96, 98), 
which is obviously true. 


(d.) In the base of a triangle find the point from which 
lines^ drawn parallel to the sides and limited by 
them^ are equal. 
If a, /3 be the sides^ any point in the base has the vector 

p = (1— ^)o+iP^. 

Per the required point 

which determines x. 

Hence the point lies on the line 

which bisects the vertical angle of the triangle. 

This is not the only solution, for we should have written 

T(l-a?)Ta= TxTfiy 

instead of the less general form above wAicA tacitly assumes that 
1 —0? and X are positive. We leave this to the student. 

{e.) Jf perpendiculars be erected outwards at the middle 
points of the sides of a triangle, each being propor- 
tional to the corresponding side, the mean point 
of the triangle formed by their extremities coincides 
with that of the original triangle. Find the ratio of 
each perpendicular to half the corresponding side of 
the old triangle that the new triangle may be equi- 

Let 2 a, 2)3, and 2(a-hj3) be the vector-sides of the triangle, 
i a unit- vector perpendicular to its plane, e the ratio in question. 
The vectors of the corners of the new triangle are (taking the 
comer opposite to 2)3 as origin) 

pj = a+eia, 

p, -2a + p + eip, 

p, = a-hi3-^i(a+)3). 


From these 

i(Pi +p.4-/>,) = i(4a + 2i3) = i(2a + 2(a + j3)), 
which proves the first part of the prppositioD. 
For the second part, we must have 

Substituting, expanding, and erasing terms common to all, the 
student will easily find 3^2 ^ j. 

Hence, if equilateral triangles be described on the sides of any 
triangle, their mean points form an equilateral triangle. 

190. Such applications of quaternions as those just made 
are of course legitimate, but they are not always profitable. In 
fact, when applied to plane problems, quaternions often de- 
generate into mere scalars, and become (§ 33) Cartesian co- 
ordinates of some kind, so that nothing is gained (though 
nothing is lost) by their use. Before leaving this class cf 
questions we take, as an additional example, the investigation 
of some properties of the ellipse. 

19L We have already seen (§31 {i)) that the equation 

p = acos0+j9sin^ 

represents an ellipse, $ being a scalar which may have any 

value. Hence, for the vector-tangent at the extremity of p 

we have dp ..,«>, 

r. =— osmd + i3cos^, 

which is easily seen to be the value of p when is increased by 


-. Thus it appears that any two values of p, for which 

differs by ~, are conjugate diameters. The area of the parallel- 

ogram circumscribed to the ellipse and touching it at the 
extremities of these diameters is, therefore, by § 96, 

4yrp^ = 4Tr(acos^-hi3sin^)(-a8ind + ^cos^) 
a constant, as is well known. 


192. For equal conjugate diameters we must have 

T(acos^+/38iii^ = y(— asm(?+j3cos(?), 
or (a«— ^«)(cos*^— 8m*fl) + 4iSaj3costfsind = 0, 

or tan2g=— ^„ ^ . 

The square of the common length of these diameters is of course 


because we see at once &om § 191 that the simi of the squares 
of conjugate diameters is constant. 

193. The maximum or minimum of p is thus found ; 
dTp_ I dp 

= — =-(-(a«-j3«)costfsin^4-'Saj3(cos»^-sin*fl)). 

For a maximum or minimum this must vanish^ hence 


tan2^ = 

i«— fl» 


and therefore the longest and shortest diameters are equally 
inclined to each of the equal conjugate diameters. Hence^ also^ 

they are at right angles to each other. 

[The student must carefully notice that here we put -j^ = 0, 

and not ^ = 0. A little reflection will show him that the 

* latter equation involves an absurdity.] 

194. Suppose for a moment a and )3 to be the greatest and 
least semidiameters. Then the equations of any two tangent- 
lines are 

p = acosB -{-fi&mO -f a?(— asind +fico8 0), 
p = acosdi-f)38in(?i+;ri(— osin(?i-h/8cos(?i). 


If these tangent lines be at right angles to each other 
5(— a8ind+)8co8^)(— asin^i+jQcostfJ = 0, 
or a* sin sin 01 +j9' cos 0008^1 =: 0. 

Also, for their point of intersection we have, by comparing 
coefficients of a, ^ in the above values of p, 

<x>s0— ;psin0 = cos^i— ^Pisin^i, 

sin0+;pcosd = BmOi-j-SiCosSi. 

Determining x^ from these equations, we easily find 

the equation of a circle ; if we take account of the above relation 
between and 0^, 

Also, as the equations above give w = — ;Pi, the tangents are 
equal multiples of the diameters parallel to them ; £k> that the 
line joining the points of contact is parallel to that joining the 
extremities of these diameters. 

195. Finally, when the tangents 

p = acos0 +j9sind +x (— asin^ -^ficoB0), 

p = acos0i+i9sindi+iri(— asin^i+jScos^i), 

meet in a given point 

p = aa+bp, 
we have 

a = cos 0^x sin = cos^i — a?i sin 0^, 

b = Bm0-\'XQOB0 = sin^i+a^iCos^i, 

Hence a?* = a* -f 5' — 1 = a?; 

and acostf + isin^ = 1 = dfcos^i+isin^j 

determine the values of and x for the directions and lengths 
of the two tangents. The equation of the chord of contact is 

p = y (a cos 4- )3 sin 0) 4- ( 1 —y) (a cos ^^ + j9 sin 0^). 

If this pass through the point 

p =zpa+qp, 

we have p ^^ yooB$+{l — y) cos O^, 

from which, by the equations which determine $ and 0^, we get 

Thus if either a and b, or p and q, be given, a linear relation 
connects the others* This, by § 30, gives all the ordinary 
properties of poles and polars. 

196. Although, in §§ 28-30, we have already given some 
of the equations of the line and plane, these were adduced 
merely for their applications to anharmonic coordinates and 
transversals; and not for investigations of a higher order. Now 
that we are prepared to determine the lengths and inclinations 
of lines we may investigate these and other similar forms 

197. The equation of the indefinite line drawn through the 
origin 0, of which the vector OA, = a, forms a part, is evidently 

or p II a, 
or Vap = 0, 
or Up^ Ua; 

the essential characteristic of these equations being that they 
are linear, and involve one indeterminate scalar in the value 
of p. 

We may put this perhaps more clearly if we take any two 
vectors, j3, y, which, along with a, form a non-coplanar system. 
Operating with S.Fafi and 8,Fay upon any of the preceding 
equations, we get 

":::) '■' = 0, ' 
and S,ayp 

Separately, these are the equations of the planes containing 
a, /3, and a^ y; together, of course, they denote the line of 



198. Conversely, to solve equations (1), or to find p in terms 
of known quantities, we see that they may be written 

8.pFap = 0,^ 

S.pray = 0,J 
so that p is perpendicular to Fafi and Fayy and is therefore 
parallel to the vector of their product. That is, 

II -a8.apy, 
or p = xa, 

199. By putting p— J9 for p we change the origin to a point 
B where OB = — /3, or BO = )3 ; so that the equation of a line 
parallel to o, and passing through the extremity of a vector fi 
drawn from the origin, is 

or p = j3-f iPcu 

Of course any two parallel lines may be represented as 
p = p +a?a, 
p = fii+af^a; 
or Fa(j)-^p) = 0, 
Fa(p^P,) = 0. 

200. The equation of a line, drawn through the extremity of 
j9, and meeting a perpendicularly, is thus found. Suppose it 
to be parallel to y, its equation is 

p = P^xy. 
To determine y we know, Jirst, that it is perpendicular to o, 
which gives Say = 0. 

Secondly, a, ^, and y are in one plane, which gives 

S.apy = 0. 
These two equations give 



whence we have 

p = p^xaFap. 

This might have been obtained in many other wajns; for 

instance^ we see at once that 

/3 = a-»a/8 = a-^SaP-^ar^Fap. 

This shows that ar^Vafi (which is evidently perpendicular to a) 
is coplanar with a and jS^ and is therefore the direction of the 
required line ; so that its equation is 

the same as before if we put — ^-r for x. 

20L By means of the last investigation we see that 

is the vector perpendicular drawn from the extremity of j3 to 
the line 

p = xa. 

Changing the origin^ we see that 


is the vector perpendicular from the extremity of j3 upon the line 

p = y+xa. 

202. The vector joining £ (where OB = j3) with any point in 
p = y-^-xa 
is y+ira— )3. 

Its length is least when 

rfr(y+a?a-i3) = 0, 
or /Sa(y + iPa— /3) = 0, 

i. e. when it is perpendicul$ur to a. 
The last equation gives 

xa^ + Saiy—p) = ^> 
or ara=— a-*/Sa(y— )3). 
T 2, 

140 QUATBRNIOKS. [ohaf. VI. 

Hence the vector peipendiculttr is 

or QT^Vaiy-p)^ ^ar^Vaffi-y), 
which agrees with the result of last section. 

SOS. To find the ahorteerfi vector distance between two Unids 

and pi=/3i+^iai; 

we must put dT{p^pi) = 0, 

or Sip^p^Xdp-dp^) = 0, 
or S{p^pi){adx+aidxi) = 0. 
Since or and Xi are independent, this breaks up into the two 
conditions Sa (p^p,) = 0, 

Saiip—pi) = ', 

proving the well-known truth that the required line is perpen- 
dicular to each of the given lines. 

Hence it is parallel to Faai, and therefore we have 

p— Pi 5= /3 + ii?«— )9i— iPjaji ^yVoai* .•. {\) 

Operate by &aoi and we get 

This determines y, and the shortest distance required is 

[Note. In the two last expressions T before 8 is simply 
inserted to ensure that the length be positive. If 

&aai(j3— )3i) be negative, 

then (§ 89) &aia(i3— )3i) is positive. 

If we omit the T we must use in the text that one of tjiese two 
expressions which is positive.] 

To find the extremities of this shortest distance, we must 

8BCT. 206.] fVHHMT Of WmtBBttIT LINS AND PLANE. 141 

operate on (I) wi<^ JS.a and S^i. We thus obtain two'equations^ 
wiiidi determine a and^lPo as y is already known. 

A somewhat different mode of treating this problem will be 
disciuwed presei^j. 

204. The equation 

5ap = 

imposes on p the sole condition of being perpendicular to a; 
and therefore^ l>eing satisfied by the veclor drawn from the 
origin to any point in a plane through the origin and perpen* 
dicular to a, is the equation of that plane. 

To find this equation by a direct process similar to that 
usually employed in coordinate geometry^ we may remark that^ 
by § 29, we may write 

where y3 and y are any two vectors perpendicular to a. In this 
form the equation contains two indeterminates, and is often 
useful ; but it is more usual to eliminate them, which may be 
done at once by operating by i&a, when we obtain the equation 
first written. 

It may also be written, by eliminating one of the ipdet^er- 
minates only, as tt^ 

where the form of the equaUoi^ shows that Skip = 0, 

205. Similarly we see that 

Saip^fi) :== 

represents a plane drawn through the extremity of fi and per- 
pendicular to <i. This, of couisse, may, like the last, be put into 
various equivalent forms. 

206. The line of intersection of the two planes 
S.a(p-p) =0,- 
and S.a. 

'(^-^) =•'''1 (1) 

142 QUATERNIONS. [chap. VL 

contains all points whose valae of p satisfies both conditions. 
But we may write (§ 92)^ since a^ a,^ and Faai are not co- 

pS.aaiVaai = Faai8,aaip+F.aiFaaiSap-j'F,F(€Mi)€LSaip, 
or, bj the given equations, 

-py« Toa, = F.a^ Faa^Safi + F. F{(M^)aSa^fi^ + x Faa^, (2) 
where x, a scalar indeterminate, is put for S.cMip which may 
have anj value. In practice, however, the two definite given 
scalar equations are generally more useM than the partially 
indeterminate vector-form which we have derived from them. 

When both planes pass through the origin we have )3 = )3i = 0^ 
and obtain at once _ xFaa 

as the equation of the line of intersection. 

207. The plane passing through the origin^ and through the 
line of intersection of the two planes (1)^ is easily seen to have 
the equation Sa,Map^SafiSa,p = 0, 

or S{aSaiP^''aiSap)p = 0. 

For this is evidently the equation of a plane passing throngh 
the origin. And, if p be such that 

Sap = Sap, 
we also have Sa^p = Sa^Pi, 

which are equations (1). 

Hence we see that the vector 

is perpendicular to the vector-line of intersection (2) of the two 
planes (1), and to every vector joining the origin with a point 
in that line. 

The student may verify these statements as an exercise. . 

208. To find the vector-perpendicular from the extremity 
of p on the plane ^ _ q 


vre must note tliat it is necessarily parallel to a, and hence that 
the corresponding value of p is 

where xa is the vector perpendicular in question. 

Hence &i(j3+a?o) = 0, 

which gives xa^ = — /Saj3, 

or xa =— a^*/Sa)3. 

Similarly the vector-perpendicular from the extremity of p on 
the plane Sa{p-y) = 

may easily be shown to be 

209. The equation of the plane which passes through the 
extremities of a, /3, y may be thus found. If p be the vector 
of any point in it, p— a, a— /3, and /3— y lie in the plane, and 
therefore (§ 101) 

5.(p-o)(a-i3)(^-y) = 0, 

or Sp( FaP + Ffiy + Fya) - S.apy = 0. 

Hence, if 8 = x{Fafi+ Fpy+ Fya) 

be the vector-perpendicular from the origin on the plane con- 
taining the extremities of a, ft y, we have 

8 = (roj3+ Ffiy-^' Fya)-'8.aPy. 

From this formula, whose interpretation is easy, many curious 
properties of a tetrahedron may be deduced by the reader. 

210. Taking any two lines whose equations are 

p = ^ -ha?a, 

p = ft + iz^iai, 
we see that S,aai{p-^b) = 

is the equation of a plane parallel to both. JFkick plane, of 
course, depends on the value of 5. 

144 QUAlTEBQillCmS. [OHIp. YI. 

Kow H istfi, the fhine contains the first fine; if b^Pt, 

the second. 

Hence^ if yFaai be the shortest vector distance between the 

lines^ we have 

S.aaAP-fii-yFaa,) = 0, 

or T{yFaa^) ^T8.ifi^p,)Uraa^, 

the result of § 203. 

21L Find the equation of the plane> passing through the 
origin^ which makes equal angles with three given lines. Also 
find the angles in question. 

Let a^ ^^ y be unit- vectors in the directions of the lines, tfnd 
let the equation of the plane be 

8bp 3= 0. 
Then we have evidently 

8aJb s Spi s 8yh = x, suppose^ 

where x 

is the sine of each of the required anglei^. 
But (§ 92) we have 

h8.aPy = x{rafi+ Vpy^ Vya). 
Hence 8.p{ Fap -f Fpy + Fya) = 

is the required equation ; and the required sine is 


212. Find the locus of the middle i)oint6 of a series of straight 
lines^ each parallel to a given plane and having its extremities 
in two fixed lines. 

Let 8yp s 

be the plane, and 

the fixed lines. Also let x and a?i correspond to the extremities 


of one of the variable lines, cr being the vector of its middle 
point. Then, obviously, 

Also Syifi—Pi-j-wa—Xia^) = 0. 

This gives a linear relation between x and a?i, so that, if we 
substitute for x^ in the preceding equation, we obtain a result 
of the form • » • ,«, 

where h and € are known vectors. The required locus is, there- 
fore, a straight line. 

213, Three planes meet in a point, and through the line of 
intersection of each pair a plane is drawn perpendicular to the 
third; prove that, in general, these planes pass through the 
same line. 

Let the point be taken as origin, and let the equations of the 

planes be 

Sap = 0, 8l3p = 0, 8yp = 0. 

The line of intersection of the first two is || Vafi, and therefore 

the normal to the first of the new planes is 


Hence the equation of this plane is 

S^pKyFap = 0, 

or SfipSay-'SapSfiy = 0, 

and those of the other two planes may be easily formed from 
this by cyclical permutation of a, /3, y. 

We see at once that any two of these equations give the 
third by addition or subtraction, which is the proof of the 

214. Given any number of points A, J5, C, &c., whose vectors 
(from the origin) are a^, aj, Oj, &c., find the plane through the 
origin for which the sum of the squares of the perpendiculars 
let fall upon it from these points is a maximum or minimum. 



Let Svp = 

be the required equation, with the condition (evidently allowable) 

?W = 1. 
The perpendiculars are (§ 208) — flr-*iS«rai, &c. 

Hence 2 S* fsra 

is a maximum. This^ves 

1.8rffa8adisr = ; 
and the condition that tr is a unit- vector gives 

Siffdw ^ 0. 
Hence, as thr may have any of an infinite number of values, 
these equations cannot be consistent unless 

where a? is a scalar. 

The values of a are known, so that if we put 

^.aSavr = (fyar, 
<^ is a given self-conjugate linear and vector function, and there- 
fore X has three values {g^g^y g^y § 164) which correspond to 
three mutually perpendicular values of -or. For one of these 
there is a maximum, for another a minimum, for the third a 
maximum-minimum, in the most general case when^i,^a,^s are 
all different. 

215. The following beautifiil problem is due to Maccullagh. 
Of a system of three rectangular vectors, passing through the 
origin, two lie on given planes, find the locus of the third. 

Let the rectangular vectors be «r, p, o-. Then by the con- 
ditions of the problem 

5Wp = 8p(r = ScT'Bj = 0, 
and Swta = 0, S^p = 0. 

The solution depends on the elimination of p and ts among these 
five equations. [This would, in general, be impossible, as p 


and -BT between them involve sijc unknown scalars ; but, as the 
tensors are (by the very form of the equations) not involved, 
the five given equations serve to eliminate the four unknown 
scalars which are really involved. Formally to complete the 
requisite number of equations we might write 

Tur = a, Tp=: b, 

but a and b may have any values whatever.] 

Prom Sa'sr = 0, fir-cr = 0, 

we have -bt = xFaa. 

Similarly, from Spp = 0, Sap = 0, 

we have p = yV^a, 

Substitute in the remaining equation 

Srffp = 0, 
and we have 

S.FaaFptr = 0, 

or SatrSpa-'-a^Safi = 0, 
the required equation. As will be seen in next Chapter, this is 
a cone of the second order whose circular sections are perpen- 
dicular to a and /3. [The disappearance of x and y in the 
elimination instructively illustrates the note above.] 

u 2, 



1. What propositions of Euclid are proved by the m&rQform 
of the equation 

p = (1— a?)o+a?j3, 

which denotes the line joining any two points in space ? 

2. Show that the chord of contact, of tangents to a parabola 
which meet at right angles^ passes through a fixed point. 

3. Prove the chief properties of the circle (as in Euclid , III) 
from the equation 

p = acos^ + i^sind; 

where Ta = 1)3, and Safi = 0. 

4. What locus is represented by the equation 

where Ta=l? 

5. What is the condition that the lines 

Top = ft Fa,p = fi„ 
intersect ? If this is not satisfied, what is the shortest distance 
between them ? 

6. Knd the equation of the plane which contains the two 
parallel lines 

Faip-^P) = 0, ra(p-ft) = 0. 

7. Find the equation of the plane which contains 

Faip^P) = 0, 
and is perpendicular to 

Syp = 0. 

8. Find the equation of a straight line passing through a 
given point, and making a given angle with a given plane. 

Hence form the general equation of a right cone. 


9. What conditions must be satisfied with regard to a number 
^of given lines in space that it may be possible to draw through a • ', 
each of them a plane in such a way that these planes may ' ^^ f 
intersect in a common line ? 

'. 10. Find the equation of the locus of a point the sum of 
the squares of whose distances from a number of given planes 
is constant. 

11. Substitute 'Mines'' for ''planes'' in (10). 

i/42. Find the equation of the plane which bisects, at right 
angles, the shortest distance between two given lines. 

Find the locus of a point in this plane which is equidistant 
fK)m the given lines. 

.13. Find the conditions that the simultaneous equations 
Sap = a, Sfip = J, Syp = (?, 
may represent a line, and not a point. 

y 14. What is represented by the equations 
iSapY = [S^pY = (%)% 
where a, )3, y are any three vectors ? 

^15. Find the equation of the plane which passes through 
two given points and makes a given angle with a given plane. 

/l6. Find the area of the triangle whose comers have the 
vectors a, )3, y. 

Hence form the equation of a circular cylinder whose axis 
and radius are given. 

^7. (Hamilton, Bishop Law^ 8 Premium Ex., 1858). 
V {a,) Assign some of the transformations of the expression 

where a and ^ are the vectors of two given points 
A and B. * 

yS.) The expression represents the vector y, or 0(7, of a 
point C in the straight line AB. 

{c.) Assign the positioin of this point C, 

^^ *v.^ :r. 5^^v ^ 

150 QUATERNIONS. [chap. VI. 

18. (Ibid.) 

(a.) If a, Pjy,h he the vectors of four points A, B, C, D, 
what is the condition for those points being in one 

(J.) When these four vectors from one origin do not thus 
terminate upon one plane^ what is the expression for 
the volume of the pyramid, of which the four points 
are the corners ? 

{c.) Express the perpendicular b let fall from the origin 
on the plane ABC, in terms of a, fi, y. 

A9. Find the locus of a point equidistant from the three planes 
Sap = 0, Spp = 0, Syp = 0. 

20. If three mutually perpendicular vectors be drawn from 
a point to a plane, the sum of the reciprocals of the squares 
of their lengths is independent of their directions. 

21. Find the general form of the equation of a plane from 
the condition (which is to be assumed as a definition) that any 
two planes intersect in a single straight line. 




4 FTER that of the plane the equations next in order 

216. -L^. Qf simplicity are those of the sphere, and of the 
cone of the second order. To these we devote a short Chapter 
as a valuable preparation for the study of surfaces of the second 
order in general. 

217. The equation 

Tp = Ta, 

or p^ = a% 

denotes that the length of p is the same as that of a given 
vector a, and therefore belongs to a sphere of radius Ta whose 
centre is the origin. In § 107 several transformations of this 
equation were obtained, some of which we will repeat here with 
their interpretations. Thus 

/SCp-f a)(p-a) = 

shows that the chords drawn from any point on the sphere to 
the extremities of a diameter (whose vectors are a and —a) are 
at right angles to each other. 

y(p-ha)(p-a) = 2TFap 

shows that the rectangle under these chords is four times the 
area of the triangle two of whose sides are a and p. 

p = (p-|-a)-*a(pH-a) (see § 105) 

shows that the angle at the centre in any circle is double that 

152 QUATERNIONS. [chap. VIL 

at the circumference standing on the same arc. All these are 
easy consequences of the processes already explained for the 
interpretation of quaternion expressions. 

218. If the centre of the sphere be at the extremity of a 
the equation may be written 

T{p^a) = Tfi, 
which is the most general form. 

If Ta = Tpy 
or a« = ^% 

in which case the origin is a point on the surface of the sphere^ 
this becomes p» - 2 iSap = 0. 

From this, in the form 

8p{p-2a) = 

another proof that the angle in a semicircle is a right angle is 
derived at once. 

219. The converse problem is — Find the locus of the feet 
of perpendiculars let fall from a given point (p = /3) on planes 
passing through the origin. 

Let Sap = 

be one of the planes, then (§ 208) the vector-perpendicular is 

—a-' Sap, 
and, for the locus of its foot, 

p = p-^a-^Sap 

[This is an example of a peculiar form in which quaternions 
sometimes give us the equation of a surface. The equation is a 
vector one, or equivalent to three scalar equations; but it 
involves the undetermined vector a in such a way as to be 
equivalent to only two indeterminates (as the tensor of a is 


evidently not involved). To put the equation in a more imme-* 
diately interpretable form, a must be eliminated, and the re- 
marks just made show this to be possible.] 

Now {p-Py = a-'S^afi, 

and (operating by 8.p) 

Adding these equations^ we get 

p'Sfip = 0, 


^\ _ mP 


so that, as is evident, the locus is the sphere of which )3 is a 

220. To find the intersection of the two spheres 

T(p-a) = Tp, 
and r(p-aO= %, 

square the equations, and subtract, and we have 

which is the equation of a plane, perpendicular to o— aj the 
vector joining the centres of the spheres. This is always a real 
plane whether the spheres intersect or not. It is, in fact, what 
is called their Radical Plane. 

221. Find the locus of a point the ratio of whose distances 
from two given points is constant. 

Let the given points be and A, the extremities of the 
vector a. Also let P be the required point in any of its 
positions, and OP z= p. 

Then, at once, if « be the ratio of the lengths of the two lines, 

This gives p*— 2iSap-ha» = «VS 



or> by an easy traiiafonnation^ 

Thus the locus is a sphere whose radius is T(- ^), and whose 

centre is at B, where OB = , a definite point in the 

1— «» 

line OA. 

222. If in any line^ OP, drawn from the origin to a given 

plane, OQ be taken such that OQ.OP is constant, find the locus 


Let Sap = a 

be the equation of the plane, 17 a vector of the required surface. 
Then, by the conditions, 

TerTp = constant = i» (suppose), 
and V^TSF = Dp. 

Irom these p = -== — = . 

Tiff -or* 

Substituting in the equation of the plane, we have 

aw' + i*iSo«r = 0, 

which shows that the locus is a sphere, the origin being situated 

on it at the point farthest from the given plane. 

223* Find the locus of points the sum of the squares of 
whose distances from a set of given points is a constant quantity. 
Find also the least value of this constant, and the correspond* 
ing locus. 

Let the vectors from the origin to the given points be Oj, Oj, 
a^ and to the sought point p, then 

-c» = {p^a,y-^{p-^a,y + +{p-CLny, 


. 2a.^_ c'-h2(a') . (2a)' 

yp-T) u — + "^' 


the equation of a sphere the vector of whose centre is — , i.e. 

whose centre is the mean of the system of given points. 

Suppose the origin to l>e placed at the mean point, the equa- 
tion becomes ^i _|. 2(a») 
^'= ^ — ' 

The right-hand side is negative^ and therefore the equation 
denotes a iipal surface^ if 

as might have been expected. When these quantities are equal, 
the locus becomes a point, viz. the origin. 

224. If we differentiate the equation 

Tp=z To. 
we get Spdp = 0. 

Hence (§ 137), p is normal to the surface at its extremity, a well- 
known property of the sphere. 

If tsr be any point in the plane which touches the sphere at 
the extremity of p, cr— p is a line in the tangent plane, and 
therefore perpendicular to p. So that 

8p{w--p) = 0, 
or i8Wp=— 2>» = a» 
is the equation of the tangent plane. 

225. If this plane pass through a given point j5, whose 
vector is )8, we have 

8pp = o». 

This is the equation of a plane, perpendicular to )8, and cutting 
from it a portion whose length is 

If this plane pass through a fixed point whose vector is y we 
must have jgpy -- a«, 


156 QUATBBNIONS. [chap. VII. 

80 that the locos of /3 is a pkne. These results contain all the 
ordinary properties of poles and polars with regard to a sphere. 

226. A line drawn parallel to y^ from the extremity of fi, 
has the equation p -. p^xy. 

This meets the sphere 

p» = o« 

in points for which x has the values given by the equation 

p* + ix8py+x^y^ = aK 

The values of x are imaginary^ that is^ there is no intersection^ if 

o«y"-H^"/3y < 0. 

The values are equals or the line touches the sphere^ if 

a«y«-|.r*^y = 0, 

or S^Py = y«03«-.a»). 

This is the equation of a cone similar and similarly situated 
to the cone of tangent-lines drawn to the sphere, but its vertex 
is at the centre. That the equation represents a cone is obvious 
from the fiict that it is homogeneous in Ty, i. e. that it is in-» 
dependent of the length of the vector y. 

[It may be remarked that from the form of the above equa- 
tion we see that, if x and of be its roots, we have 

(^7V)(^2V) = a»-^>, 
which is Euclid, III, 35, 36.] 

227* ^nd the locus of the foot of the perpendicular let fall 
from a given point of a sphere on any tangent-plane. 

Taking the centre as origin, the equation of any tangent- 
plane may be written 

iSWp = o*. 

The perpendicular must be parallel to p, so that, if we suppose 
it drawn from the extremity of a (which is a point on the sphere) 
we have as one value of «r 


'From these equations^ with the help of that of the sphere 

we must eliminate p and ^. 

We have by operating on the vector equation by S.w 
©•' = 8a'UT-\'x8vTp 

•• tsr — a a'C'Bj— a) 
Hence p = = — -^ — o— • 

Taking the tensors^ we have 

the required equation. It may be put in the form 

and the interpretation of this gives at once a characteristic 
property of the surface formed by the rotation of the Cardioid 
about its axis of symmetry. 

228, We have seen that a sphere, referred to any point what-* 
ever as origin, has the equation 

T{p^a) = 2)3. 

Hence, to find the rectangle under the segments of a chord 
drawn through any point, we must put 

where y is any unit- vector whatever. This gives 

a?V — 2a?/Say + a* = /3% 
and the product of the two values of x is 
i3'— o» 

This is positive, or the vector-chords are drawn in the same 
direction, if j^ < jli, 

i. e. if the origin is outside the sphere. 


229. AfBwte fixed points ; and, being the origin and P 
a point in space, 

find the locus of P, and explain the result when LAOB ia a 
right, or an obtuse, angle. 

Let 62 = a, OB = j3, OP = p, then 

or p*^2S{a+fi)p = -(o«+i8»), 

or r{p-(a+^)} = V{-28ap). 
While Soft is negative, that is, while LAOB is acute, the locus 
is a sphere whose centre has the vector a+jS. It Safi = 0, or 

LAOB = -, the locus is reduced to the point 

p = a+i8. 


If Z JO^ >- there is no point which satisfies the conditions. 

230. Describe a sphere, with its centre in a given line, so 
as to pass through a given point and touch a given plane. 

Let a?a, where a? is an undetermined scalar, be the vector 
of the centre, r the radius of the sphere, ft the vector of the 
giv^ point, and Syp =z a 

the equation of the given plane. 

The vector perpendicular from the point xa on the given 
plane is (§208) {a--xSya)Y-\ 

Hence, to determine x we have the equation 

T.(a-a?/Sya)>r* = T(xa--^) = r, 

so that there are, in general, two solutions. It will be a good 
exercise for the student to find from this equation the condition 
that there may be no solution, or two coincident ones. 

23L Describe a sphere whose centre is in a given line, and 
which passes through two given points. 


Let the vector of the centre be a? a, as in last section, and let 
the vectors of the points be p and y. Then, at once. 

Here there is but one sphere, except in the particular case when 
we have Ty-Tft, and Say - Sa^, 

in which case there is an infinite number. 

The student should carefully compare the results of this 
section and the last, so as to discover why in general two 
solutions are possible in the one case, and only one in the 

232. A sphere touches each of two straight lines, which do 
not meet : find the locus of its centre. 

We may take the origin at the middle point of the shortest 
distance (§ 203) between the given lines, and their equations 
wiUthenbe p = a+^^, 

where we have, of course, 

5iij8 = 0, /Sa/3i = 0. 

Let <r be the vector of the centre, p that of any point, of one 
of the spheres, and r its radius ; its equation is 

y(p-(r) = r. 
Since the two given lines are tangents, the following equations 
in X and x^ must have pairs of equal roots, 
jr(a-f arj3— a) = r. 

The equality of the roots in each gives us the conditions 

Eliminating r we obtain 

which is the equation of the required locus« 

160 QUATEENIONS. [chap. VIL 

[As we have not^ so far, entered on the consideration of the 
quaternion form of the equations of the various surfaces of the 
second order, we may translate this into Cartesian coordinates 
to find its meaning. If we take coordinate axes of x, y, z 
respectively parallel to )3, fi^, a, it becomes at once 

{x-k-myy—iy-k-mxy ^pzy 

where m and p are constants ; and shows that the locus is a 
hyperbolic paraboloid. Such transformations^ which are exceed^ 
ingly simple in all cases, will be of frequent use to the student 
who is proficient in Cartesian geometry, in the early stages of 
his study of quaternions. As he acquires a practical knowledge 
of the new calculus, the need of such assistance will gradually 
cease to be felt.] 

Simple as the above solution is, quaternions enable us to give 
one vastly simpler. For the problem^ may be thus stated-^ 
Find the locus of the point whose distances from two given 
lines are equal. And, with the above notation, the equality of 
the perpendiculars is expressed (§ 201) by 

yr.(o-(r) up = rr.(o+ (t) up^^ 

which is easily seen to be equivalent to the equation obtained 

233« Two spheres being given, show that spheres which cut 
them at given angles cut at right angles another fixed sphere. 

If c be the distance between the centres of two spheres whose 
radii are a and d, the cosine of the angle of intersection is 
evidently a^^i^^c^ 

Hence, if a, Oi, and p be the vectors of the centres, and a, a^, r 
the radii, of the two fixed, and of one of the variable, spheres; 
A and A^ the angles of intersection, we have 

(p— a)' + a'-f ^' = 2arcoBAj 
(p— Oi)*+aJ-f r* = 2airco8Ai. 


Elinunating the first power of r, we evidently must obtain a 
result such as (^ _py + J> + r» = 0, 

where (by what precedes) p is the vector of the centre, and 
fi the radius^ of a fixed sphere 

(p-^)» + i» = 0, 

which is cut at right angles by all the varjring spheres. By 
efiTecting the elimination exactly we easily find b and ft in terms 
of given quantities. 

234. To inscribe in a given sphere a closed polygon^ plane 
or gauche^ whose sides shall be parallel respectively to each 
of a series of given viectors. 

Let 2J) = 1 

be the sphere^ a, p, y, , rj, the vectors^ n in number^ and 

let pi, p^y p^ be the vector-radii drawn to the angles of. the 


Then pt—Pi = ^i^* &©•> &c. 

From this, by operating by ^^.(pa -f pi), we get 

p\^p\ = = i&p, + &pi. 

Also = Vap^—Vapx. 

Adding, we get 

= aps+JTopi = apa-j-pitt. 


— 1/ 

Pa=— a-^picu 

[This might have been written down at once from the result of 
§ 106.] 

Similarly p, = — j8-»p,/3 = ^^a'^piop, &c. 

Thus, finally, since the polygon is closed, 

P«+i = Pi = (-)*a-'ij-» p-'a-'p^ap 71$. 

We may suppose the tensors of a, j3 ly, ^ to be each unity. 

Hence, if a = a/3 rjO, 


162 QITATEBtf IONS. [ohap. VIL 

we have a-* = ^*iy-' /3-'a~S 

which is a known quaternion; and thus our condition becomee 

This divides itself into two cases^ according as n is an eyen 
or an odd number. 
If « be even^ we have 

«Pi = Pi«. 
Removing the common part p^Sa^ we have 

This gives one determinate direction^ + Va, for pi ; and shows 
that there are two^ and only two^ solutions. 
If » be odd, we have 

which requires that we have 


Hence Sap^^ = 0, 

and therefore p^ may be drawn to any point in the great circle 

of the unit-sphere whose poles are on the vector a. 

235. To illustrate these results, let us take first the case of 
« = 3. Here we must have 

S.afiy = 0, 
or the three given vectors must (as is obvious on other grounds) 
be parallel to one plane. Here apy, which lies in this plane, 
is (§ 106) the vector-tangent at the first comer of each of the 
inscribed triangles ; and is obviously perpendicular to the vector 
drawn from the centre to that comer. 
If « = 4, we have 


as might have been at once seen from § 106. 

236. Hamilton has given {Lectures, p. 674) an ingenious and 
simple process by which the above investigation is rendered 


applicable to the more difficult problem in which each side of 
the inscribed polygon is to pass through a given point instead 
of being parallel to a given line. His process depends upon the 
integration of a liuear equation in finite differences. By an 
immediate application of the linear and vector Amotion of 
Chapter V, the above solutions may be at once extended to 
any central surface of the second order. 

237. The equation of a cone of revolution, whose vertex 
is the origin, is easily found. 

Suppose a, where fa = 1, to be its axis, and e the cosine of 
its semi- vertical angle ; then, if /> be the vector of any point 
in the cone, 8aUp = z^e, 

or S*ap = —e*p*. 


238. Change the origin to the point in the axis whose vector 
is ;ra, and the equation becomes 

{—w + SatsrY = — «» (aro^-t!^)^ 
Let the radius of the section of the coue made by 
retain a constant value 6j while x changes ; this necessitates 

= e. 

Vb^ + x^ 

so that when x is infinite, e is unity. In this case the equation 
becomes ^,„^^^,^.ga ^ 0, 

which must therefore be the equation of a circular cyliuder of 
radius b, whose axis is the vector a. To verify this we have 
only to notice that if cr be the vector of a point of such a 
cylinder we must (§ 201) have 

TFaw = b, 

which is the isame equation as that aboye. 

Y 2 


289. To find, generaUy^ the equation of a cone which has 
circular sections : — 

Take the origin as vertex, and let one of the circular sections 
be the intersection of the plane 

8ap=^ 1 
with the sphere (passing through the origin) 

p« = Spp. 

These equations may be written thus, 

SaUp = -^, 

-Tp = SfiUp. 

Hence, eliminating Tp, we find the following equation which 

Up must satisfy — 

SallpSpUp =-1, 

or p*—SapSpp = 0, 
which is therefore the required equation of the cone. 

As a and fi are similarly involved, the mere /arm of this 
equation proves the existence of the subcontrary section dis- 
covered by Apollonius. 

240. The equation just obtained may be written 

or, since a and j3 are perpendicular to the cyclic arcs (§ 69*), 

sinjosiny= constant, 

where p and y are arcs drawn from any point of a spherical 
conic perpendicular to the cyclic arcs. This is a well-known 
property of such curves. 

24L If we cut the cyclic cone by any plane passing through 
the origin, as Syp = 0, 

then Fay and Ffiy are the traces on the cyclic planes, so that 
p = xUFay+yUFpy (§ 29). 


Substitute in the equation of the cone^ and we get 

where P is a known scalar. Hence the values of x and y are 
the same pair of numbers. This is a very elementary proof 
of the proposition in § 69*, that PL = MQ (in the last figure 
of that section). 

242. When x and y are equals the transversal arc becomes 
a tangent to the spherical conic, aild is evidently bisected at the 
point of contact. Here we have 

This is the equation of the cone whose sides are perpendiculars 
(through the origin) to the planes which touch the cyclic cone. 

243. It may be well to observe that the property of the 
Stereographic 'projection of the sphere, viz. that the projection 
of a circle is a circle, is an immediate consequence of the above 
form of the equation of a cyclic cone, 

244. That § 239 gives the most general form of the equation 
of a cone of the second order, when the vertex is taken as origin, 
follows from the early results of next Chapter. For it is shown 
in § 249 that the equation of a cone of the second order can 
always be put in the form 

2'2,.8ap8^p^Ap^ = 0. 
This may be written 8p<i>p = 0, 
where ^ is the self-conjugate linear and vector function 

<^p = 2r.apj3+(^ + 25ai3)p. 
By § 168 this may be transformed to 

and the general equation of the cone becomes 
(P'^S\fi)p^-\.28\p3fip = 0, 
which is the form obtained in § 239» 


245. Taking the form 

8p<l>p = 
as the simplest^ we find by differentiation 
Sdp<l>p+8pd<f>p = 0, 
or 28dp^p = 0. 

Hence ^p is perpendicular to the tangent-plane at the extremity 
of p. The equation of this plane is therefore (^ being the 
vector of any point in it) 

or^ by the equation of the oone^ 

flW^p = 0. 

246. The equation of the cone of normals to the tangent- 
planes of the given cone can be easily formed from that of the 
cone itself. For we may write it in the form 

«(*"'#)# = 0, 
and if we put ^p = o-^ a vector of the new cone^ the equation 
becomes Saifr^tT = 0. 

Numerous curious properties of these connected cones^ and of 
the corresponding spherical conies^ follow at once from these 
equations. But we must leave them to the reader. 

247. As a final example^ let us find the equation of a cyclic 
cone when five of its vector-sides are given — ^i. e. find the cone 
of the second order whose vertex is the origin, and on whose 
surface lie the vectors a, j3, y, 5> €. 

If we write 

= S.r{rapn€)F(FpyF€p)r(FybVpa), (1) 

we have the equation of a cone whose vertex is the origin — ^for 
the equation is not altered by putting xp for p. Also it is 
the equation of a cone of the second degree, since p occurs only 
twice. Moreover the vectors o, fi, y, 5, c are sides of the cone, 


because if any one of them be put for p the equation is satisfied. 
Thus if we put fi for p the equation becomes 

= S.FiFapn^) FiFfiyFtP) F(FybFpa) 

The first term vanishes because 

and the second because 

S.F^aF^yF^fi := 0, 

since the three vectors F^a, F^y, Fcft being each at right 
angles to fiy must be in one plane. 

As is remarked by Hamilton^ this is a very simple proof of 
Fascal^s Theorem — ^for (1) is the condition that the intersections 
of the planes of a^)3 and d^c; fi,y and €,p; y,h and p^a; shall 
lie in one plane ; or^ making the statement for any plane section 
of the cone, the points of intersection of the three pairs of 
opposite sides, of a hexagon inscribed in a conic, lie in one 
straight line. 


/^ 1 . On the vector of a point P in the plane 

Sap = I 

a point Q is taken, such that QO.OP is constant; find the 
equation of the locus of Q. 

2. What spheres cut the loci of P and Q in (1) so that both 
lines of intersection lie on a cone whose vertex is ? 

3. A sphere touches a fixed plane, and cuts a fixed sphere. 
If the point of contact with the plane be given, the plane of the 
intersection of the spheres contains a fixed line. 


Find the locos of the centre of the variable sphere^ if the 
plane of its intersection with the fixed sphere passes through a 
given point. 

4. Find the radii of the spheres which touchy simultaneonsly, 
the four given planes 

iSap = 0, SppszO, 8yp = 0, 58p=l. 
[What is the volume of the tetrahedron enclosed hj these 

5. If a moveable line^ passing through the origin, make with 
any number of fixed lines angles 0, 0^, 0^, &c«^ such that 

acos.d-h«iCOsA + = constant, 

where a, a^, are constant scalars, the line describes a right 


6. Determine the conditions that 

may represent a right cone. 

7. What property of a cone (or of a spherical conic) is given 
directly by the following form of its equation, 

8.1PKP = ? 

8. What are the conditions that the surfaces represented by 

8p(ftp = 0, and Aipxp = 0, 
may degenerate into pairs of planes ? 

9. Find the locus of the vertices of all right cones which have 
a common ellipse as base. 

10. Two right circular cones have their axes parallel, show 
that the orthogonal projection of their curve of intersection on 
the plane containing their axes is a parabola. 

11. Two spheres being given in magnitude and position, 
every sphere which intersects them in given angles will touch 
two other fixed spheres and cut another at right angles. 


1 2. K a sphere be placed on a table^ the breadth of the elliptic 
shadow formed by rays diverging from a fixed point is inde- 
pendent of the position of the sphere. 

13. Form the equation of the cylinder which has a given 
circular section^ and a given axis. Find the direction of the 
normal to the subcontrary section. 

14. Given the base of a spherical triangle, and the product 
of the cosines of the sides, the locus of the vertex is a spherical 
conic, the poles of whose cyclic arcs are the extremities of the 
given base. 

15. (Hamilton, Bishop Lavfs Premium Ex., 1858.) 

{a.) What property of a sphero-conic is most immediately 
indicated by the equation 

a p 
(J.) The equation 

(rAp)« + (%>)» = 

also represents a cone of the second order; X is a focal 
line, and pi is perpendicular to the director-plane cor- 

(c.) What properly of a sphero-conic does the equation 
most immediately indicate ? 

16. Show that the areas of all triangles, bounded by a 
tangent to a spherical conic and the cyclic arcs, are equal 

17. Show that the locus of a point, the sum of whose arcual 
distances &om two given points on a sphere is constant, is a 
spherical conic. 

18. If two tangent planes be drawn to a cyclic cone, the 
four lines in which they intersect the cyclic planes are sides 
of a right cone. 



19. Find the equation of the cone whose sides are the in- 
tersections of pairs of perpendicular tangent planes to a giyen 
cyclic cone. 

20. Find the condition that five given points may lie on 
a sphere. 

21. What is the surface denoted by the equation 

where p = xa-^yfi-^zy, 

a, p, y being g^ven vectors, and x^ y, z variable scalars ? 

Express the equation of the surface in terms of py a, p, y 

22. Find the equation of the cone whose sides bisect the 
angles between a fixed line and any line, in a given plane, 
which meets the fixed line. 

What property of a spherical conic is most directly given 
by this result ? 



248. npHE general scalar equation of the second order 
-*• in a vector p must evidently contain a term in- 
dependent of p, terms of the form SMpb involving p to the 
first degree, and others of the form 8.apbpc involving p to the 
second degree, «, b, c, &c. being constant quaternions. Now the 
term 8.apb may be written 


or 8a8prb^8b8pra^8.pfbray 

each of which may evidently be put in the form 8yp, where 
y is a known vector. 

Similarly the term 8.apbpc may be reduced to a set of terms, 
each of which has one of the forms 

Ap\ {8apy, Sap8pp, 

the second being merely a particular case of the third. Thus 

(the numerical factors 2 being introduced for convenience) we 

may write the general scalar equation of the second degree as 

follows : — 

22./Sap/Sl3p+^p'+2%= C. (1) 

249. Change the origin to D where OD = 5, then p becomes 
p-^by and the equation takes the form 

22.8ap8pp-^Ap^ + 2:^{8ap8fib+8pp8ab)'^2A8dp + 28yp 

+ 22.8ab8pb+Ab'-i'28yb-C=z 0; 

from which the first power of p disappears, that is tie surface 
is referred to its centre, if 

^{a8p»-^p8ab)i-Ab + y = 0, (2) 


a vector equation of the first degree^ which in general gives 
a single definite value for b, by the processes of Chapter V. 
[It would lead us beyond the limits of an elementary treatise 
to consider the special cases in which (2) represents a line, 
or a plane, any point of which is a centre of the surface. The 
processes to be employed in such special cases have been amply 
illustrated in the Chapter referred to.] 
With this value of b, and putting 

the equation becomes 

22.&vfi)3p+^« = i). 

If i^=0, the surface is conical (a case treated in last Chapter); 
if not, it is an ellipsoid or hyperboloid. Unless expressly stated 
not to be, the surface will, when D is not zero, be considered 
an ellipsoid. By this we avoid for the time some rather delicate 

By dividing by D, and thus altering only the tensors of the 
constants, we see that the equation of central surfaces of the 
second order, referred to the centre, is (excluding cones) 

21{8apSfip)+ffp' =^ I (3) 

250. Differentiating, we obtain 

22{8adp8pp + 8apSpdp} -^2ff8pdp = 0, 
or 8.dp{^{a8pp-^p8ap)+ffp} = 0, 

and therefore, by § 137, the tangent plane is 

8{vr^p){l{a8pp + p8ap)+ffp} = 0, 
i.e. 8.^{2{a8fip+p8ap)-h^p} = 1, by (3). 

Hence if v = 2{a8pp-\-p8ap)+ffp (4) 

the tangent plane is 81;^ = 1, 

and the surface itself is 8vp = 1. 

And, as y-* is evidently the vector-perpendicular from the origin 
on the tangent plane, v is called the vector of proximity. 


25L Hamilton uses for v, which is obviously a linear and 
vector function of p, the notation (pp, (p expressing a functional 
operation^ as in Chapter V. But, for the sake of clearness, we 
will go over part of the ground again, especiallj for the benefit 
of students who have mastered only the more elementary parts 
of that Chapter. 

We have, then. 

With this definition of ^, it is easy to see that 

(a.) ^(p-f-cr) = <l>p + <l><T, &c., for any two or more vectors. 
(6.) <p{xp) = X(f>py a particular case of (a), x being a scalar. 
{c.) d<l>p = 4>{dp). 

(d.) Scrimp = 1 {Sa<TSpp + Sfi(r Sap) -^ff Spar = Sptpar, 
or <^ is, in this case, self-conjugate. 
This last property is of great importance. 

252. Thus the general equation of central surfaces of the 
second degree (excluding cones) may now be written 

Sp<l>p^l (1) 

Differentiating, Sdp<f>p +'Spd<t>p = 0, 

which, by applying (c.) and then {d.) to the last term on the 

left, gives 2S<t>pdp=. 0, 

and therefore, as in § 250, though now much more simply, the 
tangent plane at the extremity of p is 

/S(tsr-p)<^p = 0, 

or SmiPp = Sp<f}p = 1. 

K this pass through A{OA = a), we have 

Sa<l>p = 1> 

or, by (d.), Spi^a = 1, 

for all possible points of contact. 


This id therefore the equation of the plane of contact of 
tangent planes drawn from A, 

253. To find the enveloping cone whose vertex is A, notice 

ttat {Sf4p^ 1) +i>(5>)<^a- \y = 0, 

where j9 is any scalar^ is the equation of a surface of the second 
order touching the ellipsoid along its intersection with the plane. 
If this pass through A we have 

andj9 is found. Then our equation becomes 

(S/M(>p-l)(/S»<^a-l)-(/S^-l)« = 0, (1) 

which is the cone required. To assure ourselves of this, transfer 
the origin to A, by putting p + a for p. The result is, using 
{a.) and {d.\ 
{Sp<l>p + 28p<l>a-i'8a4>a-l){Sa4>a^l)-{8p4>a-i'Sa4>a- 1)» = 0, 
or Sp<l)p{Sa(t>a — 1 ) - {Sp<t>ay = 0, 

which is homogeneous in Tp^, and is therefore the equation of 
a cone. 

Suppose A infinitely distant, then we may put in (1) ^a for a, 
where x is infinitely great, and, omitting all but the higher 
terms, the equation of the cylinder formed by tangent lines 
parallel to a is 

(SJj^p- 1) Sa<l)a-{Sp<l>ay = 0. 

254. To study the nature of the surface more closely, let us 
find the locus of the middle points of a system of parallel chords. 

Let them be parallel to a, then, if «r be the vector of the 
middle point of one of them, 'ur-^-xa and 'ur-^xa are simultaneous 
values of p which ought to satisfy (1) of § 252. 

That is 8.{^±xa)4>{:fir±xa) = 1. 

Hence, by {a.) and (rf.), as before, 

Sm(fyGr + x^ Sa<t>a =1, 

Sw<t>a=0 (1) 


The latter equation shows that the locus of the extremity 
of 'Gj^ the middle point of a chord parallel to a^ is a plane 
through the centre, whose normal is ^a; that is, a plane 
parallel to the tangent plane at the point where OA cuts the 
surface. And (d.) shows that this relation is reciprocal — so that 
if 13 he any value of -or, i.e. be any vector in the plane (1), o 
will be a vector in a diametral plane which bisects all chords 
parallel to ^. The equations of these planes are 

Sxa^a = 0, 

&fs^^ = 0, 
so that if F.ffxuf)^ = y (suppose) is their line of intersection, we 

^^^^ Sycl>a = = 8a4>y ' 

Sy(l>fi =z = Sp(l>Y ►, (2) 

and (1) gives Sp(l>a = = iSo<^/3 . 

Hence there is an infinite number of sets of three vectors 
<^y Py yy ^^^ ^^^ ^^^ chords parallel to any one are bisected 
by the diametral plane containing the other two, 

255. It is evident from § 23 that any vector may be ex- 
pressed as a linear function of any three others not in the 

same plane, let then 


where, by last section, 

8a<i>p = S^(i>a = 0, 

Sai^y =: Sy<l)a =^ 0, 

8fi(l>y = 8y(l>fi = 0. 

And let . iSou^a = 1 

8^P = 1 

Sy(l>y = 1 J 

SO that a, /3, and y are vector conjugate semi-diameters of the 
surface we are engaged on. 

Substituting the above value of p in the equation of the 


sur&oe^ and attending to the equations in.a,Pfy and to {a.), {b.), 
and (d.), we have 

To transform this equation to Cartesian coordinates^ we notice 
that X is the ratio which the projection of p on a bears to a 
itself^ &c. If therefore we take the conjugate diameters as 
axes of ^, fi, Cf <uid their lengths as a, b, c, the above equation 
becomes at once 

^. + i. + ^. - ^> 

the ordinary equation of the ellipsoid referred to conjugate 

256. If we write — ^* instead of ^, these equations assume 
an interesting form. We take for granted^ what we shall after- 
wards prove^ that this halving or extracting the root of the 
vector function is lawful^ and that the new linear and vector 
function has the same properties {a.\ (b.), {c.)^ {d.) (§ 251) as 
the old* The equation of the surface now becomes 


or S'^p^p = — 1, 

or, finally, 7\^p = 1. 

If we compare this with the equation of the unit-sphere 

we see at once the analogy between the two sur&ces. The 
^here can be changed into the ellipsoid, or vice versa, by a linear 
deformation of each vector, the operator being the function \^ or its 
inverse. See the Chapter on Physical Applications. 

257. Equations (2) § 254 now become 

iSa\^«/3 = = SyjfaylfP, fee, (1) 

so that ^a, ^)3, yfry, the vectors of the unit^sphere which cor- 


respond to aemUconjtigate diameters of the ellipsoidy form a rect- 
angular system. 

We may remark here, that, as the equation of the ellipsoid 
referred to its principal axes is a case of § 256, we may now 
suppose i,y, and k to have these directions, and the equation is 

J?* y* ^» _ . _ . 

"Y + ^ ,+ ^ = 1> which, m quaternions, is 

a* d' c* 

We here tacitly assume the~ existence of such axes, but in all 
cases, by the help of Hamilton's method, developed in Chapter V, 
we at once arrive at the cubic equation which gives them. 
It is evident from the last- written equation that 
A.. . *^^*^ _^j8jp kSkp 

and ^p^^C^ +-^^ + ^, 

^ a b c ^ 

which latter may be easily proved by showing that 

And this expression enables us to verify the assertion of last 
section about the properties of V^. 

As Sip = — a?, &c., X, y, z being the Cartesian coordinates 
referred to the principal axes, we have now the means of at once 
transforming any quaternion result connected with the ellipsoid 
into the prdinary one. 

258. Before proceeding to other forms of the equation of the 
ellipsoid, we may use those already given in solving a few 

Find the loeus of a point when the perpendicular Jrom the centre 
on its polar plane is of constant length. 

If «r be the vector of the point, the polar plane is 
iS|p<^'cr =1, 

and the length of the perpendicular from is ijrrz (§ 205). 

A a 


Hence the required locus is 

or Svr<l>*vr^—C*, 

a concentric ellipsoidj with its axes in the same direction as 
those of the first. By § 257 its Cartesian equation is 

^* + 5* + ^i - ^ • 

258. M^ the locus of a point whose distcmcefrom a given point 
is ahoays in a given ratio to its distance front a given line. 

Let /j=a?)3 be the given line, and A (OA = a) the given point, 
and let Sa^ = 0. Then for any one of the required points 

T{p^a) = eTrfip, 
a surface of the second order, which may be written 
p«-2iSap+a« = e^{8^pp-^PY). 
Let the centre be at d, and make it the origin, then 
p' + 28p{b^a)+{b-ay = e' {S\p{p+b)^P'{p + by], 
and, that the first power of p may disappear, 

(d-a) = ^»(/3/Sl35-i3^8), 
a linear equation for S. To solve it, note that Sa^ = 0, operate 
by 8.p and we get 

(1 -6«/3« + 6«/3»)S>3« = Spb = 0. 
Hence 5 — a = — e^fi% 

or 8 = 


Referred to this point as origin the equation becomes 

which shows that it belongs to a surface of revolution whose 
axis is parallel to ^, as its intersection with a plane Spp = a, 
perpendicular to that axis, lies also on the sphere 


260. A sphere, passing through the centre of an ellipsoid^ is 
cut by a series of spheres whose centres are on the ellipsoid and 
which pass through the centre thereof $ Jmd the envelop of the 
planes of intersection. 

Let (/}— o)* = a' be the first sphere, i.e. 

p«-2^ap = 0. 
One of the others is 

• p*— 2i8Wp=0, 

where &Gr<^^ 1. 

The plane of intersection is 

8{'ar^a)p = 0. 

Hence, for the envelop, (see next Chapter,) 


' f where m^z 


0, i 

or ^ = 

xp, {Vx^ 

= 0}, 

i. e. tsr = 



a?«6'/)0-V = 




and, eliminating x, 

Spi^'-p = 


a cone 

of the second order. 

26L From a point in the outer of two concentric ellipsoids a 
tangent cone is drawn to the inner, find the envelop of the plane of 

If Sur<pw=z 1 be the outer, and 8p\lrp = 1 be the inner, 4> and 

yjr being any two self-conjugate linear and vector functions, the 

plane of contact is 

Svnjrp^ 1. 

Hence, for the envelop, Sv/yjrp = 0, "i 

iSW'^ = 0, J 

therefore (fm = xyjrp, 

or -or = Xfpr^y^p, 

A a 2 


This g^ves xS.y^pf^-^y^p = 1^ > 

and x*S.y^p4>''hlrp = 1, ) 

and thereforej eliminating x, 

8.ylfp<t>-'y^p = 1, 
or S,p\lf<l>-*\lfp = 1, 

another concentric ellipsoid^ as ^<^-^^ is a linear and vector 
Amotion = x suppose ; so that the equatioi^ may be written 

Spxp = 1. 

262. Find tie locus of intersection of tangent planes at tie 
extremities of conjugate diameters, 

U a, fi,y be the vector semi-diameters^ the planes are 

Swyjf^a = — 1, 

with the conditions § 257. 

— ^S.^a^^^i/ry = ^ = ^a+^I^P-^ylryj by § 92, 
therefore ^-bt = \/3, 

since yjfay ^j9, ^y form a rectangular system of unit- vectors. 
This may also evidently be written 
Smjf^'Gr =—3, 
showing that the locus is similar and similarly situated to the 
given ellipsoid, but larger in the ratio \/3 : 1. 

263. Find the locus of the intersection of three sphere whm 
diameters are semi-conjugate diameters of an ellipsoid. 

If a be one of the semi-conjugate diameters 

And the corresponding sphere is 

p'— /Sap = 0, 
or p" — iiS\/ra\/f-*p = 0, 



with similar equations in p and y. Henoe^ by § 92^ 
and^ taking tensors^ 

or JV^-V"' = v^3, 

or, finally, Sp^~*P = — 3 p*. 

This is Fresnel^s Surface of Elasticity in the Undulatory Theory. 

264. Before going farther we may prove some useful proper- 
ties of the function ^ in the form we are at present using — viz. 
iSip jSjp kSkp 

We have p = ^iSip^jSjp^kSkp, 

and it is evident that 

i ^- J ,7 * 


_ ^'^^P J'^P ^^^P 
^-»p = aH8ip+b^J8;p-^e*kSkp, 

and so on. 

Again, if a, jS, y be any rectangular imit- vectors 

oa9«= ^, +-^ 

-r ' — —} 

&c. = &c. 

But as 

(5ip)' + (^p)' + (%)'=-p', 

we have 

Sa>t>a+SP4,p+Sy4^ = J. + J_ + i.. 

8.ii^P4>y = 



a* ' 

a» ' 

a" ' 

b^ ' c« 

b- ' c- 

Sjy Sky 
b* ' c^ 

— 1 
"■ a^b^c^ 

Sia, Sja, Ska 
Sip, Sjp, Skp 
Siy, Sjy, Sky 



And so on. These elementary investigations are given here for 
the benefit of those who have not read Chapter Y. The student 
may easily obtain all such results in a far more simple maimer 
by means of the formula of that Chapter. 

265. Find the locus of intersection of a rectangular system of 
three tangents to an ellipsoid. 

K cr be the vector of the point of intersection, o, ft y the 
tangents, then, since v-\-xa should give equal values of or when 
substituted in the equation of the surface, giving 
i8'(or4-^a)^(«r4'iPa) = 1, 
or x^8a4>a + 2x8ml>a+ (5W^— 1) = 0, 

we have (5W^)* = &i^(5Br<^— 1). 

Adding this to the two similar equations in ft and y 
{8a4^y + iSfi<hry + (Sy<K)* = iSo4>a + 8p<l>p + Syi^y) {STfft^^ 1), 

or -(^)« = (^ + 1- + ^) (&r<^-l), 

an ellipsoid concentric with the first. 

266. If a rectangular system of chords be drattm through any 
point within an ellipsoid^ the sum of the reciprocals of the rect- 
angles under the segments into which they a/re divided is constant. 

With the notation of the solution of the preceding problem, 
«r giving the intersection of the vectors, it is evident that the 
product of the values of x is one of the rectangles in question 
taken negatively. 

Hence the required sum is 

_L i- i. 
2&w^a g« "^ g' "^ C 

Smiftrff — 1 S'sr^'sr — 1 

This evidently depends on Server only and not on the par- 


ticular directions of a,p,y: and is therefore unaltered if 'cf be 
the vector of any point of an ellipsoid similar^ and similarly 
situated, to the given one. [The expression is interpretable 
even if the point be exterior to the ellipsoid.] 

267. Show that if any rectangular system of three vectors he 
drawn from a point of an ellipsoid, the plane containing their 
other extremities passes through a fixed point. Find the locus of 
the latter point as the former varies. 

With the same notation as before, we have 

SxiTifyGr = 1, 

and 8{sr-^xa)(l>{vr+xa) = 1 ; 

therefore a? = ^ , . 


Hence the required plane passes through the extremity of 

and those of two other vectors similarly determined. It therefore 
passes through the point whose vector is 

aSa(f>zr + pSp(lm + ySytfm 

(9 = tsr— 2 

Swl>a-\'Spit>p + SY(t>y 

or (9 = tsr+ ^^ (§ 173). 

m^ ^^ ^ 

Thus the first part of the proposition is proved. 
But we have also 


whence by the equation of the ellipsoid 
the equation of a concentric ellipsoid. 


268. Find tie directions of the three vectors wAicA are parallel 
to a set of conjugate diameters in eacA of two central surfaces of 
tAe second degree^ 

Transferring the centres of both to the origin, let their 
equations be 

8(4p = 1 or 0, ^ .J. 

and Spyirp = 1 or 0. i ^ 

If o, /3, y be vectors in the required directions, we must have 

8a(l>p = 0, Sayjffi = 0, ^ 

8p<l>y^0, Spylry==0, ^ (2) 

Syifya = 0, iSy^o = 0. ^ 

From these equations 

<l>a\\rpyUa, &c. 
Hence the three required directions are the roots of 

r.<^/Wrp = (3) 

This is evident on other grounds, for it means that if one of 
the surfaces expand or contract uniformly till it meets the other , 
it will touch it successively at points on the three sought vectors. 

We may put (3) in either of the following forms — 

or^^ii^p = 0,J 

and, as ^ and i/r are given functions, we find the solutions by 
the processes of Chapter V. 

[Note. As <fr^y\r and ^-*0 are not, in general, self-conjugate 
functions, equations (4) do not signify that a, ft y are vectors 
parallel to the principal axes of the surfaces 

-^#=1, 5.pV^-^^p=l.] 

269* Find the equation of the ellipsoid of which three con- 
jugate semi-diameters are given. 


Let the vector semi-diameters be a, p, y, and let 
8p<l>p s= 1 
be the equation of the ellipsoid. Then (§ 255) we have 
ScuJHi = 1, Sa4>p = 0, 

Syijyy = 1, Sy<l>a = ; 

the six scalar conditions requisite (§ 139) for the determination 
of the linear and vector function 0. 

They give a || r<^/3</)y, 

or xa = <l>~^ Fpy» 
Hence x = xScuIm = S.afiy, 

and similarly for the other combinations. Thus^ as we have 

pS.apy = aS.pyp-^, 
we find at once 

4>pS*.aPy = FpyS.pyp+ VyaS.yap-^' FafiS.oifip ; 
and the required equation may be put in the form 
SKapy = SKapp'\-SKfiyp + S*.yap. 

The immediate interpretation is that if four tetrahedra he formed 
hy grottping, three and three, a set of semi-conjugate vector axes 
of an ellipsoid and any other vector of the surface, the sum of 
the squares of the volumes of three of these tetrahedra is equal to 
the square of the volume of the fourth. 

270. When the equation of a surface of the second order can 

be put in the form 

^p</>-V=l, (1) 

where (</>-y) (</>— ^i) (</>— y>) = 0, 

we know that g, g^^g^ are the squares of the principal semi- 
diameters. Hence, if we put </>-|-A for </> we have a second 
surface, the differences of the squares of whose principal semi- 
axes are the same as for the first. That is, 

^p(*-h*)-V=l (2) 



is a surface canfocal with (1). From this simple modification 
of the equation all the properties of a series of confocal surfaces 
may easily be deduced. We give one as an example. 

27L Any two confocal suffaces of the second order , whkb meet, 
intersect at right angles. 

For the normal to (2) is, evidently, 

and that to another of the series, if it passes through the 
common point whose vector is p, is there 

= j^^ «p((0+ao-*-(«+A)-Op> 

and this evidently vanishes if h and h^ are difiPerent, as they 
must be unless the surfaces are identical. 

272. To find the conditions of similarity of two central surfaces 
of the second order, 

Referring them to their centres, let their equaitions be i 

^^P-'>\ ^1) 

Now the obvious conditions are that the axes of the one are 
proportional to those of the other. Hence, if 

g' -^m^g^ + mig '{■m^0,'\ 

/' + </" + </+»*'= 0, J 

be the equations for determining the squares of the reciprocals 

of the semiaxes, we must have 

-:t- = M> -z- = M% — = f^> v3) 

m^ Ml m 

|vbere fi is an undetermined scalar* Thus it appears that there 


:l\ ■■ ™ 


are but two scalar conditions necessary. Eliminating fx we 

flll\ M^ VIM2 fill 

which are equivalent to the ordinary conditions. 

273. Find the greatest and least semi-diameters of a central 
plane section of an ellipsoid. 

Here iSp</>p = 1 


together represent the elliptic section; and our additional coU"> 
dition is that ^ is a maximum or minimum. 

Differentiating the equations of the ellipse, we have 

/S^prfp = 0, 

Sddp = 0, 

and the maximum condition gives 

dTp = 0, 

or Spdp = 0. 

Eliminating the indeterminate vector dp we have 

8,ap4>p=iO (2) 

' This shows that tAe mammum or minimum vector, the normal at 
its extremity y and the perpendicular to the plane of section, lie in 
one plane. It also shows that there are but two vectors which 
satisfy the conditions, and that they are perpendicular to each 
other, for (2) is satisfied if ap be substituted for p. 

We have now to solve the three equations (1) and (2), to find 
the vectors of the two (four) points in which the ellipse (1) 
intersects the cone (2). We obtain at once 

^p =a xF.ffr^aFap. 

Operating by S.p we have 

1 = xp^8a^-^a. 
B b 2 

188 QlTATB]tNIOK& [chap.VIII. 

Hence p.^p^^p^a^l^ 

fromwhich &o(l-p*^)-*o = 0; (4) 

a quadratic equation in p*, from which the lengths of the 
maximum and minimum vectors are to be determined. By 
§ 147 it may be written 

«p*&i^-»o-p«-S.o(«,-*)o+a« = (5) 

[If we had operated by S.<l>-^a instead of by S.a, we should 
have obtained an equation apparently different from this^ but 
easily reducible to it. To prove their identity is a good exercise 
for the student.] 

Substituting the values of p* given by (5) in (3) we obtain 
the versors of the required diameters. [The student may easily 
prove directly that 

(l-pl^)-»a and (l-pj*)-*a 

are necessarily perpendicular to each other^ if both be perpen- 
dicular to a, and if pj and pi be different.] 

274. By (4) of last section we see that 

Hence the area of the ellipse (1) is 


Also the locus of normals to all diametral sections of an ellipsoid, 
whose areas are equal, is the cone 

8(j4-^a = (7o». 
When the roots of (4) are equal, i. e. when 

(««,a*-^5a0a)^ = 4 ma*Sa(l>'% (5) 

the section is a circle. It is not di£Scult to prove that this 


equation is satisfied by only two values of Ua, but another 
quaternion form of the equation gives the solution of this and 
similar problems by inspection. 

275. By § 168 we may write the equation 

8p<t>p = 1 
in the new form 

8.kpfip-\-pp* = 1, 

where /? is a known scalar^ and X and fx are definitely known 
(with the exception of their tensors^ whose product alone is 
given) in terms of the constants involved in (f>, [The reader is 
referred again also to §§ 121^ 122.] This may be written 

28KpSixp'{-{p^8KiJL)p^ = 1 (1) 

From this form it is obvious that the surface is cut by any 
plane perpendicular to X or m in a circle, For^ if we put 

8\p = a, 
we have 2a8p.p'{'{p^8\p)p^ = 1, 

the equation of a sphere which passes through the plane curve 
of intersection. 

Hence X and fi of § 168 are^ the values of a in equation (5) of 
last section. 

276. -4«y two circular sections of a central surface of the second 
order J whose planes a/re not parallel^ lie on a sphere. 

For the equation 

{8Kp—a){8p.p'^h) = 0, 
where a and h are any scalar constants whatever^ is that of a 
system of two non-parallel planes^ cutting the surface in circles. 
Eliminating the product 8Kp8[kp between this and equation (1) 
of last section^ there remains the equation of a sphere. 

277. To find the generating lines of a central surface of the 
second order, 

. Let the equation be 8p<\>p = 1 ; 


then^ if a be the vector of any })omt on the surface^ and 9 
a vector parallel to a generating line^ we must have 

p =: a+xvr 
for all values of the scalar x. 

Hence iS(o+a?cr)^(a-f^ = 1, 

which gives the two equations 

Saifyer = 0, ^ 
Sm<l>vr = 0. J 
The first is the equation of a plane through the origin parallel 
to the tangent plane at the extremity of a, the second is the 
equation of the asymptotic cone. The generating lines are 
therefore parallel to the intersections of these two surfaces^ as is 
well known. 

From these equations we have 

where ^ is a scalar to be determined. Operating on this by 
&j9 and S.y, where /3 and y are any two vectors not coplanar 
with a, we have 

StsT(y(t>P+ Vap) = 0, Sm{y<l>y- Vya) = (1) 

Hence^iyi^fi + ?^a/3)0</)y- Vya) = 0, 

or my^S.aPy-'ScupaS.afiy = 0. 

Thus we have the two values 

belonging to the two generating lines. 

278. But by equation (1) we have 

Zfsr = F:(y</)^+ raP){y4>y- Tya) 

= my^<t>-^rpy+yF.<l>arpy''aS.aTpY; 
which^ according to the sign of y, gives one or other generating 

Here Fpy may be any vector whatever, provided it is not 


perpendicular to a (a condition assumed in last section)^ and 
we may write for it $. 

Substituting the value of y before founds we have 

= r.<t>ar.o4-'0±^^2^r,l>ae. 



or, as we may evidently write it^ 

^(tr'{<l>ae)±y/^F4>a0 (2) 

Put r = r<i>ae, 

and we have ,8^ 

ziff = dr^Var + </ — ^— r, 

with the condition 8r^ = 0. 

[Any one of these sets of values forms the complete solution of 
the problem; but more than one have been given^ on account 
of their singular nature and the many properties of surfaces 
of the second order which immediately follow from them. It 
will be excellent practice for the student to show that 

yffO = U{r4ara<i>''0 ± ^ ^ V^ai) 

is an invariant. This may most easily be done by proving that 

V.y^Q^By^ = identically.] 

Perhaps^ however, it is simpler to write a for F./Sy, and we thus 

2ftsr = </)-' r,a ra4>a ± ^ — ^ Vw^a, 

[The reader need hardly be reminded that we are dealing 
with the general equation of the central surfaces of the second 
order — ^the centre being origin.] 

192 QUATBRNIOira [cHAP.Vni. 

!• Find the locus of points on the surface 

where the generating lines are at right angles to one another. 

2. Find the equation of the surface described by a straight 
line which revolves about an axis^ which it does not meet but^ 
with which it is rigidly connected. 

3. Find the conditions that 

may be a sur&ce of revolution. 

4. Find the equations of the right cylinders which dream- 
scribe a given ellipsoid. 

5. Find the equation of the locus of the extremities of per- 
pendiculars to plane sections of an ellipsoid^ erected at the 
centre^ their lengths being the principal semi-axes of the sec- 
tions. [Fresnd^s Wave'Surface.'] 

6. The cone touching central plane sections of an ellipsoid^ 
which are of equal area, is asymptotic to a confocal hyperboIoid« 

7. Find the envelop of all non-central plane sections of an 
ellipsoid whose area is constant. 

8. Find the locus of the intersection of three planes, per- 
pendicular to each other, and touching, respectivdy, each of 
three confocal surfaces of the second order. 

9. Find the locus of the foot of the perpendicular £rom the 
centre of an ellipsoid upon the plane passing through the ex- 
tremities of a set of conjugate diameters. 


10. Find the points in an ellipsoid where the inclination of 
the normal to the radius- vector is greatest. 

11. If four similar and similarly situated surfaces of the 
second order intersect, the planes of intersection of each pair 
pass through a common point. 

12. If a parallelepiped be inscribed in a central surface of 
the second degree its edges are parallel to a system of conjugate 

13. Show that there is an infinite number of sets of axes 
for which the Cartesian equation of an ellipsoid becomes 

14. Find the equation of the surface of the second order 
which circumscribes a given tetrahedron so that the tangent 
plane at each angular point is parallel to the opposite face ; and 
show that its centre is the mean point of the tetrahedron. 

15. Two similar and similarly situated surfaces of the second 
order intersect in a plane curve, whose plane is conjugate to the 
vector joining their centres. 

16. Find the locus of all points on 

SfKpp = 1, 

where the normals meet the normal at a given point. 

Also the locus of points on the surface, the normals at which 
meet a given line in space. 

17. Normals drawn at points situated on a generating line 
are parallel to a fixed plane. 

18. Find th^ envelop of the planes of contact of tangent 
planes drawn to an ellipsoid from points of a concentric sphere. 
Find the locus of the point from which the tangent planes are 
drawn if the envelop of the planes of contact is a sphere. 

19. The sum of the reciprocals of the squares of the per- 
pendiculars from the centre upon three conjugate tangent planes 
is constant. 

c c 


20. Cones are drawn^ touching an ellipsoid^ from any two 
points of a similar^ similarly situated^ and concentric ellipsoid* 
Show that they intersect in two plane curves. 

Find the locus of the vertices of the cones when these plane 
sections are at right angles to one another. 

21. Find the locus of the points of contact of tangent planes 
which are equidistant from the centre of a surface of the second 

22. From a fixed point A, on the surface of a given sphere, 
draw any chord AD; let 1/ be the second point of intersection 
of the sphere with the secant BD drawn from any point B; and 
take a radius vector AE, equal in length to BI/, and in direc- 
tion either coincident with^ or opposite to, the chord AB : the 
locus of ^ is an ellipsoid, whose centre is A, and which passes 
through B. (Hamilton, Elements, p. 227.) 

23. Show that the equation 

l*{e*^\){e'\'8aa) = (&ip)«-2<?iSiip-8'aV-f(&i»>-|-(l-dV. 

where e is a variable (scalar) parameter, represents a system of 
confocal surfaces. {Ibid, p. 644.) 

24. Show that the locus of the diameters of 

which are parallel to the chords bisected by the tangent planes 
to the cone 

Spy^p = 
is the cone 

-8'.p0t^-*<^p = 0. 

25. Find the equation of a cone, whose vertex is one summit 
of a given tetrahedron, and which passes through the circle 
circumscribing the opposite side. 

26. Show that the locus of points on the surface 

= 1, 


the normals at wliich meet that drawn at the point p = «r^ is 
on the cone 

27. Find the equation of the locus of a point the square of 
whose distance from a given line is proportional to its distance 
from a given plane. 

28. Sho^ that the locus of the pole of the plane 

Sap=^ 1, 
wiUi respect to the surface 

8p4>p = 1, 
is a sphere^ if a be subject to the condition 

29. Show that the equation of the surface generated by lines 
drawn through the origin parallel to the normals to 

Spip'^p = 1 

along its lines of intersection with 

Sp{<l> + i)-'p = 1, 

is «r«—M«r (</>+*)-* tsr = 0. 

30. Common tangent planes are drawn to 

2S\p8iip-^{P'-8Xp)p^ = 1, and Tp = A, 

find the value of A that the lines of contact with the former 
surface may be plane curves. What are they^ in this case^ on 
the sphere ? 

31. If tangent cones be drawn to 

Sp^P = 1, 
from every point of 

8p(l>p = 1, 

the envelop of their planes of contact is 
8p<l>'^p = 1. 
c c a 

196 QUATERNIONS, [chap. VIII. 

32. Tangent cones are drawn from every point of 

5(p— a)^(p— o) = «S 
to the similar and similarly situated surface 

Sp<l>p = 1, 
show that their planes of contact envelop the surface 
(5o<^p— !)• =in^8f4p. 


33. Find the envelop of planes which touch the parabolas 

ps=at*+pt, p^zar^-^-yr, 
where a, p, y form a rectangtdar system^ and t and r are scalais. 

34. Find the equation of the surface on which lie the lines of 
contact of tangent cones drawn from a fixed point to a series of 
similar^ similarly situated^ and concentric ellipsoids. 

35. Discuss the surfaces whose equations are 

SapSpp = Syp, 
and S^ = 1. 

36. Show that the locus of the vertices of the right cones 
which touch an ellipsoid is a hyperbola. 



279. We have already seen (§ 31 (l)) that the equations 

p = </)^=2.a/(0, 
and p = <l>{t, u) = ^.af{ti u), 

where a represents one of a set of given vectors, and/ a scalar 
function of scalars t and u, represent respectively a curve and 
a surface. We commence the present brief Chapter with a few 
of the immediate deductions from these forms of expression. 
We shall then give a number of examples, with little attempt 
at systematic development or even arrangement. 

280. What may be denoted by t and u in these equations 
is, of course, quite immaterial : but in the case of curves, con- 
sidered geometrically, t is most conveniently taken as the length, 
s, of the curve, measured from some fixed point. In the Kine- 
matical investigations of the next Chapter t may, with great 
convenience, be employed to denote time. 

28L Thus we may write the equation of any curve in space as 

where <^ is a vector ftmction of the length, *, of the curve. Of 
course it is only a linear frmction when the equation (as in 
§ 31 (A)) represents a straight line. 

282. We have also seen (§§ 38, 39) that 


is a vector of unit length in the direction of the tangent at the 
extremity of p. 

At the proximate point, denoted hy s+is, this onit tangent 
vector becomes ^'*-|.<^''*d*+&c. 

But, because y^'^ - 1^ 

we have S^if/s^t'^s sz O, 

Hence (f/^s is a vector in the oscukting phme of the curve, and 
perp^idicukr to the tangent. 

Also, if M be the angle between the successive tangents 4>* 
and ip^^-^if/'ibS'^ , we have 

•cl = ^*"'> 

so that tie temar of ^'s is the reciprocal of tie radius ofalsohU 
curvature at tie point s. 

288. Thus, if OP =r ^« be the vector of any point P of the 
curve, and if C be the centre of curvature at P, we have 

and thus OC = ^« 77- 

<!> s 

is the equation of the locus of the centre of curvature. 

Hence also Kif/sif/'s or <^'#</)"# 

is the vector perpendicular to the osculating plane ; and 

is the tortuosity of the given curve, or the rate of rotation of its 
osculating plane per unit of length. 

284. As an example of the use of these expressions let us 
Jind tie curve wiose curvature and tortuosity are both constant. 

We have 

curvature = T^''s :sz Tp'z=i c. 


Hence <^'«<^''* = //p"= co, 

where a is a unit vector perpendicular to the osculating plane. 

This gives 

if c^ represent the tortuosity. 

Integrating we get 

p'ff'^Cp'^p, (1) 

where j8 is a constant vector. Squaring both sides of this 
equation, we get 

or T^ = \/c«+<?;. 
Multiply (1) by p', remembering that 

Tp'= 1, 
and we obtain ^p'z=, —Ci +p'j3, 

or p'= c^s—pP + a, (2) 

where a is a constant quaternion. Eliminating p\ we have 

of which the vector part is 

p"-.p/3» = -c^sp'^ Fap. 
The complete integral of this equation is evidently 

p = fcos,*y^+t;sin.*y^-^(e?»*/3-hrai3), (3) 

^ and rj being any two constant vectors. We have also by (2), 

Spp = CiS-{-8a, 
which requires that 

Spi = 0, S^ri = 0. 

The farther test, that 

^'= 1, gives us 

- 1 = r^«(f»sin».«IS3 + t;»cos».«T^- 2-S^sin.*T^cos.*2)3) ^ 

c> + ej 


This requires^ of course^ 

^-«> ^=^=3^ 

so that (3) becomes the general equation of a helix traced on 
a right cylinder. (Compare § 31 {m).) 

285. The vector perpendicular from the origin on the tan- 
gent to the curve p = 4>8 

is, of course, ^ f' ^, /r^ / 

-rfpp, or pFpp 

(since p' is"a unit vector). 

7b find a common property of curves whose tangents are all 
equidistant from the origin. 

Here TVpp = c, 

which may be written 

^p^^S*pp'=c* (1) 

This equation shows that, as is otherwise evident, every curve 

on a sphere whose centre is the origin satisfies the condition. 

For obviously 

— p* = c" gives 8pp'z=, 0, 

and these satisfy (1). 

If 8pp' does not vanish, the integral of (1) is 

^Tp^-c^ = s, (2) 

an arbitrary constant not being necessary, as we may measore 
s from any point of the curve. The equation of an involute 
which commences at this assumed point is 

-BJ = p — *p'. 
This gives JW« = Tp^ + «« - 2s8pp' 

= Tp^+s^^2s^/T^^^^\ by (1), 
= c\ by (2). 

This includes all curves whose involutes lie on a sphere about tie 


286. Mnd the locus of tie foot of the perpendicular drawn to a 
tangent to a right helix from a point in the axis. 

The equation of the helix is 

s ^ . s 
p = acos- H-jSsin- -\-ys, 
a a ' 

where the vectors a, )3, y are at right angles to each other^ and 

Ta = 2)3 = h, while aTy = ^a'^-h^. 

The equation of the required locus is^ by last section, 

isr = pTp/ 

/ s tf«— i« . *v ^/ . « a»— J* s^ . *• 

= a(cos-H ;— *sin-)+i3(sm — *cos-)+ y-r*. 

^ a a" a^ ^^ a a* a/ ' a^ 

This curve lies on the hyperboloid whose equation is 

S^am^-S^^—a^S^ym = **, 

as the reader may easily prove for himself. 

287. To find the least distance between consecutive tangents to a 
tortuous curve. 

Let one tangent be 

tsr = p-\-Xffy 

then a consecutive one, at a distance Is along the curve, is 

The magnitude of the least distance between these lines is, by 
§§ 203, 210, 

Trp'p"h» ' 

•if we neglect terms of higher orders. 



It may be written^ since pp" is a vector, and 7/= 1, 


But ^^^^y - r''^>'";u - ^ n'^.vv- 

Hence ^^s.Up^^FpY 

UVpY Vpr 


is the small angle, d^, between the two successive positions of 
the osculating plane. [See also § 283.] 

Thus the shortest distance between two consecutive tangents 
is expressed by the formula 

12r ' 
where r, = ^tttj is the radius of absolute curvature of the tor- 

tuous curve. 

288. Let us recur for a moment to the equation of the 
parabola (§ 31 (/)) 


Here p'= {a+^f)-^. 

whence, if we assume Sa^ = 0, 

d» I 

T- = V— o*— /3'^», 

firom which the length of the arc of the curve can be derived in 
terms of t by integration. 

Again, p"= («+^0 -^ +fi{-^)'' 

But ^-± 1 _ , ^ S.p{a+^ 

d»* ~ ds' T{a+pi) ~ ^ ds T{a+^()' ' 


and therefore, for the vector of the centre of cterrature we have 
(§ 283), 

which is the quaternion equation of the evolute. 

289. One of the simplest forms of the equation of a tortuous 
curve is 

where a, )8, y are any three non-coplanar vectors, and the 
numerical factors are introduced for convenience. This curve 
lies on a parabolic cylinder whose generating lines are parallel 
to y; and also on cylinders whose bases are a cubical and a 
semi-cubical parabola, their generating lines being parallel to 
j3 and a respectively. We have by the equation of the curve 

yP V dt 

,'=(.+,<+ i-)^. 

from which, by Tp=^ 1, the length of the curve can be found 
in terms of ^j and 

from which p" can be expressed in terms of *. The investiga- 
tion of various properties of this cuirve is very easy, and will 
be of great use to the student. 

It is to be observed that in this equation t cannot sfemd for 
*, the length of the curve. Such an equation as 

or even the simpler form 

p = cu?+^*% 
involves an absurdity. 

1} da 

204 QUATEBKION& {chap. H. 

290. The equation _ ^i^ 

where ^ is a self-conjugate linear and vector fonetion^ t sl scalar 
yariable, and c an arbitrary vector constant^ denotes a curious 
class of curves. 
We have at once 

where \og4> is another self*conjugate linear and vector fiinction, 
which we may denote by x* These functions are obviously 
commutative^ as they have the same principal set of rectangular 
vectors, hence we may write 

which of course gives 
As a verification, we should have 

= (i+8^x + Ux*+ )p 

where e is the base of Napier's Logarithms. 
This is obviously true if 

<^« = e^x, 
or = ex, 

or log<^ = x, 
which is our assumption. 

[The above process is, at first sight, rather startling, but the 
student may easily verify it by writing, in accordance with the 
results of Chapter V, 

whence <f>U = — /la/Sae— ^,0/S/3€— ^iyiSye. 


He will find at once 

and the results just given follow immediately.] 
29L That the equation 

represents a surface is obvious from the fact that it becomes 
the equation of a definite curve whenever eitAer t or u has a 
particular value assigned to it. Hence the equation at once 
fiimishes us with two systems of curves, lying wholly on the 
surface, and such that one of each system can, in general, be 
drawn through any assigned point on the surface. Tangents 
drawn to these curves at their point of intersection must, of 
course, lie in the tangent plane, whose equation we have thus 
the means of forming. 

292. By the equation we have 

where the brackets are inserted to show that these quantities 
are partial differential coefficients. If we write this as 

dp = <l>t dl + <f>u du^ 
the normal to the tangent plane is evidently 

and the equation of that plane 

293. As a simple example, suppose a straight line to move 
along a fixed straight line, remaining always perpendicular to it, 
while rotating about it through an angle proportional to the 
space it has advanced; the equation of the ruled surface de- 
scribed will evidently be 

/) = o/ + wOcos^-f ysin^), (1) 


where a, fi,y Bie rectangular vectors, and 

This surface evidently intersects the right cylinder 

p = a(j8cos^+y8in^)-f ra, 
in a helix (§§ 31 {m), 284) whose equation is 
p = o^-|-a(j8cos^-|-ysin^). 

These equations illustrate very well the remarks made in §§ 31 
{l)f 291 J as to the curves or sur&ces represented by a vector 
equation according as it contains one or two scalar variables. 
Prom (1) we have 

dp = [o— «()8sin^— ycos^)]rf^+(/3cos^-fysin^rf«, 
so that the normal at the extremity of p is 

To (y cos ^— /3 sin ^)-«Jj3» J/o. 
Hence, as we proceed along a generating line of the surface, 
for which t is constant, we see that the direction of the normal 
changes. This, of course, proves that the surface is not de- 

284. Hence the criterion for a developable surface is that if 
it be expressed by an equation of the form 

p = 4>t + uylrt, 

where ^ and yjrt are vector functions, we must have the direction 

of the normal 

independent of u. 

This requires either Fyjrt^^t =zO, 

which would reduce the surface to a cylinder, all the generating 
lines being parallel to each other ; or 
r<l/tylrtz=z 0. 
This is the criterion we seek, and it shows that we may write, 
for a developable surface in general, the equation 

p = 0^-f«^7 (1) 


Evidently p =z <^t 

is a curve (generally tortuous) and <^'^ is a tangent vector. 
Hence a developable surface is the locus of all tangent lines to a 
tortuous curve. 

Of course the tangent plane to the surface is the osculating 
plane at the corresponding point of the curve; and this is 
indicated by the fact that the normal to (1) is parallel to 
Vi^'t^"t (See § 283.) 

295. A Geodetic line is a curve drawn on a surface so that its 
osculating plane at any point contains the normal to the surface. 
Hence, if i; be the normal at the extremity of p, p and p" the 
first and second differentials of the vector of the geodetic, 

which may be easily transformed into 
r.vdUp'^ 0. 

296« In the sphere JJ) = a we have 

hence S.ppp''^ 0, 

which shows of course that p is confined to a plane passing 
through the origin, the centre of the sphere. 
For a formal proof, we may proceed as follows — 

The above equation is equivalent to the three 

Sep = 0, S0p = 0, S0p'-O, 

from which we see at once that ^ is a constant vector, and 
therefore the first expression, which includes the others, is the 
complete integral. 

Or we may proceed thus — 

= -p5.ppV'+/>"&pV= r.rppTpp''= F.Fpp'dFpp', 
whence by § 133 (2) we have at once 

UFpp^=z const. = suppose, 
which gives the same results as before. 

208 QUATBKNIONS. [chap. IX. 

287* Ib aDj cone we have^ of course^ 
8vp = 0, 
since p lies in the tangent plane. But we have also 

8vp = 0. 
Hence^ by the general equation of § 295^ eliminating v we get 
= (hS.pp'Fpy= SpdUp by § 133 (2). 

C^SpUp'-^jsdpUp'^z SpUp' -^JTdp. 

The interpretation of this is^ that the length of any arc of the 
geodetic is equal to the projection of the side of the cone (drawn 
to its extremity) upon the tangent to the geodetic. In other 
wordsj when the cane is developed the geodetic becomes a straight 
line. A similar result may easily be obtained for the geodetic 
lines on any developable surface whatever. 

298. To find the shortest line connecting two points on a given 

Here / Tdp is to be a minimum^ subject to the condition that 
dp lies in the given surface. 

Now hJTdp = jhTdp = -/^^^ = -fs.Udpddp 

= -[5. Udp 5p] +J8.bp d Udp, 

where the term in brackets vanishes at the limits^ as the 
extreme points are fixed^ and therefore bp = 0. 
Hence our only conditions are 


' S.bpdUdp = 0, and Svbp = 0, giving 
V.vdUdp = 0, as in § 295. 
If the extremities of the curve are not given, but are to lie 


on given curves, we most refer to the integrated portion of 
the expression for the variation of the length of the arc. And 

^*s^<>™ S.Udphp 

shows that the shortest line cuts each of the given curves at 
right angles. 

299. The osculating plane of the curve 

is 8.<f/t<l/'t{^'-p) = 0, (1) 

and is^ of course^ the tangent plane to the surface 

p = (l>t+u(l/t : (2) 

Let us attempt the converse of the process we have, so far, 
pursued, and endeavour to find (2) as the envelop of the variable 
plane (1). 

Differentiating (1) with respect to t only, we have 
&0V(tir-p) = 0. 
By this equation, combined with (1), we have 

or -orsp+iw^'sr 0+«^^'> 

which is equation (2). 

300. This leads us to the consideration of envelops generally, 
and the process just employed may easily be extended to the 
problem (yi finding the envelop of a series of surfaces whose equa- 
tion contains one scalar parameter. 

When the given equation is a scalar one, the process of 
finding the envelop is precisely the same as that employed in 
ordinary Cartesian geometry, though the work is often shorter 
and simpler. 

If the equation be given in the form 

p = yjf{t, «, r), 
E e 

210 QUATBRKIONS. [cttA]». IX. 

where ^ is a Teetor fimctioili i and « the scalar Yariables for 
any one sorfiu^^ 9 the soaku* parameter^ we have for a proximate 

Hence at all points on the intersection of two successiye sm&ces 
of the series we have 

which is equivalent to the following scalar equation connecting 
the quantities t, n, and v ; 

This equation, along with 

enables us to eliminate t, u, p, and the resulting scalar equation 
is that of the required envelop. 

SOL As an example, let us find the envelop of the osculating 
plane of a tortuous curve. Here the equation of the plane is 
(§299), 5.(tir-p)<^7<^''^ = 0, 

or «r = 4>t'^x<l/t'^y4rt = ^(ir,y, f), 

if p zs (t>t 

be the equation of the curve. 
Our condition is, by last seotion, 

or S4't 4/'t lift'\^xi^''t+y^'''f\ = 0, 

or y84t^"i^"H^ 0. 

Now the scalar factor cannot vanish, unless the given curve 
be plane, so that y = 

and the envelop is 

the developable surface, of which the given curve is the edge 
of regression, as in § 299. 


302. When the equation contains two scalar parameters its 
differential coefficients with respect to them must vanish^ and 
we have thus three equations from which to eliminate two 
numerical quantities. 

A very common form in which these two parameters appear 
in quaternions is that of an unknown unit- vector. In this case 
the problem may be thus stated — Find the envelop ofihe mrface 
whose scalar expiation is 

where a is subject to the one condition 

Ta s 1. 
JDifferentiating with respect to a alone^ we have 
Svda = 0, Soda = 0, 

where r is a known vector frmction of p and a. Since da may 
have any of an infinite number of values^ these equations show 

*^* Vav = 0. 

This is equivalent to two scalar conditions only, and these, 
in addition to the two given scalar equations, enable \ui to 
eliminate a. 

With the brief explanation we have given, and the examples 
which follow, the student will easily see how to deal with any 
other set of data he may meet with in a question of envelops. 

303. Find the envelop of a plane whose distance from the origin 
is constant 

Here Sap^-^c, 

with the condition Ta =z I. 

Hence, by last section, 

Fpa = 0, 

and therefore p = ^a, 

or Tp = c, 

the sphere of radius c, as was to be expected. 

212 QUATEENIONS. [chap. IX. 

If we seek the envelop of thoee only of the planei which are 
parallel to a given vector fi, we have the additional relation 


In this case the three differentiated equations are 
Spda = 0, Sada = 0, Spda = 0, 
and they give S.afip = 0. 

Hence a=:U.pFpp, 

and the envelop is TFpp = cTp, 

the circular cylinder of radius c and axis coinciding with ^. 

By putting Safi = e, where ^ is a constant different from zero^ 
we pick out all the planes of the series which have a definite 
inclination to fi, and of course get as their envelop a right cone. 

304. The equation 

8*ap + = * 
represents a parabolic cylinder, whose generating lines are 
parallel to the vector aFafi, For it is not altered by increasing 
p by^he vector xaFafi; also it cuts planes perpendicular to a 
in one line, and planes perpendicular to Fafi in two parallel lines. 
Its form and position of course depend upon the values of a, A 
and b. It is required to find its envelop if fi and b be constant, 
and a be subject to the one scalar condition 

Ta= 1. 
The process of § 302 gives, by inspection, 

p/Sap+ Ffip = xa. 
Operating by &a, we get 

S^ap-^-S.aPp =— ar, 
which gives S.afip = ar + i. 

But, by operating successively by S, Ffip and by &/), we have 
{FfipY = xSmPp, 
and {p^—x)Sap = 0, 


Omitting^ for the present^ the factor Sap, these three equations 
give, by elimination of a? and a, 

which is the equation of the envelop required. 

This is evidently a surface of revolution of the fourth order 
whose axis is fi ; but, to get a clearer idea of its nature, put 

<?» p-* = tsr, 

and the equation becomes 

which is obviously a surface of revolution of the second degree, 
referred to its centre. Hence the required envelop- is the 
reciprocal of such a surface, in the sense that the rectcmgle under 
the lengths of condirectional radii of the two is constant 

We have a curious particular case if the constants are so 
related that j+iS* = 

for then the envelop breaks up into the two equal spheres, 
touching each other at the origin, 

p« = ± Sfip, 

while the corresponding surface of the second order becomes 
the two parallel planes 

305. The particular solution above met with, viz. 
8ap = 0, 

limits the original problem, which now becomes one of finrling 
the envelop of a line instead of a surface. In fact this equation, 
taken in conjunction with that of the parabolic cylinder, belongs 
to that generating line of the cylinder which is the locus of the 
vertices of the principal parabolic sections. 
Our equations become 

Sap = 0, 


2b =1; 
whence Vfip = a?o, giving 

X ^ '■^8,ctpp ^ — — > 
and thenoe * TFfip = -; 

so that the envelop is a circular cylinder whose axis is ^. [It 
is to be remarked that the equations above require that 

Safi = 0, 
so that the problem now solved is merely that of the envelop of 
a parabolic cylinder wAicA rotates about its focal line. This dis- 
cussion has only been entered into for the sake of explaining a 
peculiarity in a former result, because of course the present 
results can be obtained immediately by an exceedingly simple 

806« The equation 

8ap8.afip = a^, 

with the condition Ta = 1, 

represents a series of hyperbolic cylinders. It is required to foul 
tAeir envelop. 
As before, we have 4- Vpp 8ap = xa, 
which by operating by S.a, and 8.Fpp, gives 

2a* =— ar, 
p*8.afip =s x8apj 
{Fppy8ap szx8.afip. 
Eliminating a and x we have, as the equation of the enve 

p"(r/3p)« = 4 a*. 
Comparing this with the equations 

and (r/3p)«=-2aS 


ivfaich represent a sphere and one of its circumscribing cylinders^ 
we see that^ if condirectional radii of the three surfaces be drawn 
from the origin, that of the new surface is a geometric mean 
between those of the two others. 

307. -K^^ the envelop of all spheres which touch one given line 
and have their centres in another. 

Let p = P+yy 

be the line touched by all the spheres, and let xa be the vector 

of the centre of any one of them, the equation is (by § 200, 

or § 201) 

y^ip^xay = --(r.y (i3-^a))% 

or^ putting for simplicity, but without loss of generality, 

IV = 1, Safi=:0, Sfiy = 0, 

so that p is the least vector distance between the given lines, 

and, finally, p*— j3*-r2a?51ap = x^S^ay, 

Hence, by § 300, — 2/Sap = 2x8*ay, 
[This gives no definite envelop if 

Say = 0, 
i. e. if the line of centres is perpendicular to the line touched by 
all the spheres.] 

Eliminating x, we have for the equation of the envelop 

iS»ap + ^"ay(p»-/3») = 0, 

which denotes a surface of revolution of the second degree, 
whose axis is a. 

Since, from the form of the equation, Tp may have any 
magnitude not less than T/3, and since the section by the plane 

is a real circle, on the sphere 

the surface is a hyperboloid of one sheet. 

216 QUATBBKIOKS. [chap. IX. 

[It will be instructive to the student to find the signs of the 
values of ^1,^,,^, as in § 165^ and thence to prove the above 

308. As a final example let us find the envelop of the hyper- 
bolic cylinder 

Sap Sfip^c = 0, 

where the vectors a and /3 are subject to the conditions 

Ta = 253 = 1, 

Say = 0^ Sfib = 0. 

[It will be easily seen that two of the six scalars involved in 
a, p still remain as variable parameters.] 

We have Soda = 0, Syda = 0, 

so that da = xFay. \ 

Similarly dp =s yFfib. 

But^ by the equation of the cylinders^ 

Sap Spdfi + Spda Sfip = 0, 

or ySap S.pbp -\-x8.ayp Spp = 0. 

Now by the nature of the given equation^ neither Sap nor Sfip 
can vanish^ so that the independence of da and dp requires . 

. S.ayp = 0, S.pbp = 0. 

Hence a = U.yFyp, p =iU,h7hp, 


and the envelop is I 

T.Vypnp-cTyb ^ 0, 

a surface of the fourth order, which may be constructed by 
laying off mean proportionals between the lengths of condirec- 
tional radii of two equal right cylinders whose axes meet in the | 

ongm. I 


309* We may now easily see the truth of the following 
general statement. 


Suppose the given equation of th^ series of surfaces, whose 
envelop is required, to contain m vector, and n scalar, para- 
meters ; and that the latter are subject to j» vector, and j scalar, 

In all there are Bm + n scalar parameters, subject to 3p-\-q 
scalar conditions. 

That there may be an envelop we must therefore in general 

*^av^ {Sm+n)-^{3p + q)=zl, or =2. 

In the former case the enveloping surface is given as the locus 
of a series of curves, in the latter a series ot points. 

Differentiation of the equations gives us 3jo-f j-f 1 equations, 
linear and homogeneous in the 3m-\-n differentials of the scalar 
parameters, so that by the elimination of these we have one 
final scalar equation in the first case, two in the second; and 
thus in each case we have just equations enough to eliminate 
all the arbitrary parameters. 

310. To find the locus of the foot of the perpendicular drawn 
from the origin to a tangent plane to any surface. 

If Svdp = 

be the differentiated equation of the surface, the equation of 
the tangent plane is 

8{iff-^p)v = 0. 

We may introduce the condition 

8vp = 1, 

which in general alters the tensor of v, so that v~^ becomes the 
required vector perpendicular, as it satisfies the equation 

Siffv = 1. 
It remains that we eliminate p between the equation of the 
given surface, and the vector equation 

tff = v~^. 

The result is the scalar equation (in w) required. 

Ff ^ 

218 QUATBRNIONS. [chap. IX. 

For exsmplej if the given surface be the ellipsoid 
SfHt>p = 1, 
we have «•""* = y = ^, 

so that the required equation is 

iS«r-»^-*«r-» = 1, 
or Sw^'^^w = «r*, 

which is Fresnel's Surface of Elasticity. (§ 263.) 

31L To find the reciprocal of a given surface with respect to 
the unit sphere whose centre is the origin. 

With the condition 

Spv = 1, 

of last section^ we see that — y is the vector of the pole of 

the tangent plane 

8{w-p)v = 0. 

Hence we must put w =—v, 

and eliminate p by the help of the equation of the given surface. 
Take the ellipsoid of last section^ and we have 
w =— <^p, 
so that the reciprocal surfiEice is represented by 
5W^~^«j = 1. 

It is obvious that the former ellipsoid can be reproduced 
£rom this by a second application of the process. 
And the property is general^ for 
Spv = 1 
givesj by differentiation^ and attention to the condition 

Svdp = 0, 
the new relation Spdv = 0^ 

so that p and v are corresponding vectors of the two surfaces : 
either being that of the pole of a tangent plane drawn at the 
extremity of the other. 


312. If the given surface be a cone with its vertex at the 
origin, we have a peculiar case. For here every tangent plane 
passes through the origin, and therefore the required locus is 
wholly at an infinite distance. The difficulty consists in Spv 
becoming in this case a numerical multiple of the quantity 
which is equated to zero in the equation of the cone, so thai 
of course we cannot put as above 

Spv = 1. 

313* The properties of the normal vector v enable us to 
write the partial differential equations of families of surfaces in 
a very simple form. 

Thus the distinguishing property of Cylinders is that all their 
generating lines are parallel. Hence all positions of v must 
be parallel to a given plane^or 

Sou = 0, 

which is the quaternion form of the well-known equation 

dF dF ^_o 
dx dy dz "^ ' 

To integrate it, remember that we have always 

8vdp = 0, 

and that as i^ is perpendicular to a it may be expressed in terms 
of any two vectors, /3 and y, each perpendicular to a. 

Hence v = xfi+yy, 

and xSpdp + ySydp = 0. 

This shows that Sfip and Syp are together constant or together 

variable, so that 

Spp =/(8yp), 

where/ is any scalar function whatever. 

314. In Surfaces of Revolution the normal intersects the axis. 
Hence, taking the origin in the axis a, we have 

220 QUATERNIONS. [cHAP. IX. = 0, 
or ' V = xa+yp. 

Hence xSadp+ySpdp = 0, 

whence the integral Tp =/{Sap). 

The more common form^ which is easily derived from that just 
written, is jfjr^^ ^ F{8ap). 

In Cone9 we have Svp = 0, 

and therefore 

Svdp = S.v{TpdUp+UpdTp) = TpSvdUp. 
Hence SvdUp = 0, 

so that V must be a function of Up^ and therefore the integral is 

nup) = 0, 

which simply expresses the fact that the equation does not 
involve the tensor of p. 

315. y equal lengths be laid off on the normals drawn to any 
surface , the new surface formed by their extremities is normal to 
the same lines. 

For we have m = p+aUp, 

and Svdw^ = Svdp+aSvdUv = 0, 

which proves the proposition. 

Take, for example, the surface 

Spfpp = 1 ; 

the above equation becomes 


so that p = (^ -f l) 'fir, 

and the equation of the new surface is to be found by eliminating 

-pjT— (written x) between the equations 


and = 8.<f>{af<t>+ l)-*tEr<^(a?<^+ l)-*tEr. 

316. It appears from last section that if one orthogonal 
surface can be drawn cutting a given system of straight lines, 
an indefinitely great number may be drawn: and that the 
portions of these lines intercepted between any two selected 
surfaces of the series are all equal. 

Let p ^ (T-f a?r, 

where <t and r are vector fiinctions of p, and x is any scalar, 
be the general equation of a system of lines : we have 

Srdp = = S(p—(r)dp 

as the dijBTerentiated equation of the series of orthogonal surfaces, 
if it exist. Hence the following problem. 

317. It is required to find the criterion of integraUlity of the 
equation g^^^ ^ ^ (^^ 

as the complete differential of the equation of a series of surfaces. 

Hamilton has given {Elements^ p. 702) an extremely elegant 
solution of this problem, by means of the properties of linear 
and vector fiinctions. We adopt a different and less simple 
process, on account of some results it offers which will be useful 
to us in the next Chapter ; and also because it will show the 
student the connection of our methods with those of ordinary 
differential equations. 

If we assume Fp ^ C 

to be the integral, and apply to it the very singular operator 
devised by Hamilton, 

_ , d , d ^ , d 

,dF ,dF ^dF 
we have VFp =^ ^ t- +Jj- + ^'r ' 


But p = iaf+J3f'\-kz, 

whence dp = idx-^jdy+kdz, 

and =dF= ^^+ ^^^^-^Tz^^ ^-^SVFdp. 

Comparing with the given equation^ we see that the latter 
represents a series of surfaces if v^ or a scalar multiple of it^ 
can be expressed as V^. 
If v = V^, 


have V. = V.i' = -(^ + ^ + ^), 

a well-known and most important expression^ to which we shall 
return in next Chapter. Meanwhile we need only remark that 
the last-written quantities are necessarily scalars, so that the 
only requisite condition of the integrability of (1) is 

rVy=0 (2) 

K V do not satisfy this criterion^ it may when multiplied by 
a scalar. Hence the farther condition 

rV{wp) = 0, 
which may be written 

FvVw+wFVv = (3) 

This requires that 

SpVv s (4) 

If then (2) be not satisfied, we must try (4). If (4) be satisfied 
w will be found from (3); and in either case (1) is at onoe 

p[f we put dv = (t>dp 

where ^ is a linear and vector fiinction, not necessarily self- 
conjugatCj we have 

by § 173. Thus, if <^ be self-conjugate, c = 0, and the criterion 


(2) is satisfied. If ^ be not self-conjugate we have by (4) for the 
criterion g^^ _ 

These results accord with Hamilton's, lately referred to, but the 
mode of 6btaiQing them is quite different from his.] 

318. As a simple example let us first take lines diverging 
from a point. Here r||p, and we see that ifv=^p 

so that (2) is satisfied. And the equation is 

Spdp = 0, 

whose integral Tp = C 

gives a series of concentric spheres. 

Lines perpendicular to, and mtersecling, a fixed line. 

If a be the fixed line, /3 any of the others, we have 

8.c^p = 0, Safi = 0, 8^dp = 0. 

Here v || a Fop, 

and therefore equal to it, because (2) is satisfied. 
Hence 8.dpaVap = 0, 

or a*8pdp^Sap8adp = 0, 

whose integral is the equation of a series of right cylinders 

319. To find tie orthogonal trajectories of a series of circles 
whose centres are in, and their planes perpendicular to, a given 

Let a be a unit- vector in the direction of the line, then one 
of the circles haa the equations 

Tp^C, ^ 
8ap= Cr,S 
where C and C are any constant scalars whatever. • 

224 QUATERNIONS. [chap. IX. 

Henoe^ for the required surfaces 

V II d^p 11 Vapy 
where d^p is an element of one of the circles, v the normal to 
the orthogonal surface. Now let dp be an element of a tangent 
to the orthogonal surface^ and we have 

Svdp = S.apdp =• 0. 

This showtf that dp is in the same plane as a and p, i. e. that 
the orthogonal surfaces are planes passing through the com- 
mon axis. 

[To integrate the equation 

S,apdp = 
evidently requires, by § 317, the introduction of a fiictor. For 
rVFap = F{irai+jraJ+irai) 
= 2a, 
so that the first criterion is not satisfied. But 
S.FapF^rap = 28.aFap = 0, 
so that the second criterion holds. It gives, by (3) of § 317, 
F.^wFap-\'2wa = 0, 
or pSaVw—aSp^w + 2wa = 0. 

That is SaVto = 0, ^ 

SpVw = 2w. J 
These equations are satisfied by 

But a simpler mode of integration is easily seen. Our equation 

may be written 

S.apdp = 0, 

0=.S.aF^ = Sa^-^^dS.alogU^^ 
p Up ^ p 

which is immediately integrable, ^ being an arbitrary but con- 
stant vector. 


As we have not introduced into this work the logarithms 
of versors, nor the corresponding angles of quaternions^ we must 
refer to Hamilton's Elements for a farther development of this 

320. To find the orthogonal trajectories of a given series of 

If the equation 

i^P = C, 

give &vdp = 0, 

the equationjf of the orthogonal curves is 

Yvdp = 0. 

This is equivalent to two scalar differential equations (§ 197)^ 
which^ when the problem is possible^ belong to surfaces on each 
of which the required lines lie. The finding of the requisite 
criterion we leave to the student. 

Let the surfaces he concentric spheres. 

Here p» = G, 

and therefore Ypd^ = 0. 

Hence dJJp = — UpVpdp = 0, 

and the integral is 

Up = constant^ 

denoting straight lines through the origin. 

Let the surfaces he spheres touching each other at a common 
paint. The equation is (§ 218) 

Sap-^ = C, 

whence V.papdp = 0. 

The integrals may be written 

S.afip = 0, p^ + hTFap = 0, 
the first (fi being any vector) is a plane through the common 
diameter; the second represents a series of rings or tores (§ 322) 

226 QUATBRNIOK& [cwitf . IX 

formed by the revolation^ about a, of circles icnehing that line 
at the point common to the spheres. 

Let the surfaces be similar, similarly siiuated, and eonceniriej 
surfaces of tie second order. 

Here Spxp = C, 

therefore ^XP^P == ^' 

But^ by § 290; the integral of this eqnation » 

p = ^xe 

where ^ and x &i^ related to each other^ as in § 290 ; and e is 
any constant vector. 

32L Find tie general equation of surfaces described by a line 
which always meets, at right angles j a fixed line, 

K a be the fixed line^ /3 and y forming with it a rectangular 

unit sjrstem^ then 

p = ara+jr(/3 + ^y), 

where y may have all values, but x and z are mutuaUy de- 
pendent, is one form of the equation. 

Another, expressing the arbitrary relation between x and z, is 

But we may also write 

as it obviously expresses the same conditions. 

The simplest case is when F{x) =s kx. The surface is one 
which cuts, in a right helix, every cylinder which has a for 
its axis. 

322. The centre of a sphere moves in a given circle, fiid the 
equation of the ring described. 

Let a be the unit-vector axis of the circle, its centre the 
origin, r its radius, a that of the sphere. 


Then (p-fiY - -«^ 

is the equation of the sphere in any position, where 
«a^ = 0, Tfi = r. 

These give S.afip = 0, and fi must now be eliminated. The 
result is that ^ - raUVap, 

giving (p« — /-a + a^'Y z= 4:r''T^ Fap, 

which is the required equation. It may easily be changed to 

(p«— ^a + r^)' =:-4eiap»— 4r»iS'»ap, (1) 

and in this form it enables us to give an immediate proof of 
the very singular property of the ring (or tore) discovered by 

For the planes 

S.p(a±— !i_) = 0, 

which together are represented by 

f^{r^^a^)8'ap^a^8^pp = 0, 
evidently pass through the origin and touch (and cut) the ring. 
The latter equation may be written 

r»S»ap-a^{8^ap + S''pUfi) = 0, 

or r^S^ap+a^ip^-^-S^apUp) :=: (2) 

The plane intersections of (1) and (2) lie obviously on the 

new surface 

(p^^a^ + f^Y = ^a^S^.apUp, 

which consists of two sphwes of radius r, as we see by writing 
its separate factors in the form 

(p±aaUpy + r'' = 0, 

323. It may be instructive to work out this problem from 
a diff^^nt point of view, especially as it affords excellent prac- 
tice in transformations. 

G g ^ 


A circle revolver about- an tuns passing within it, tie perpen- 
dicular from the centre on the axis lyifig in the plane of the circle : 
show that, for a certain position of the axis, the same solid may 
be traced out by a circle revolving about an external axis in its 
own plane. 

Let a = ^/4^+? be the radius of the circle, i the vector axis 
of rotation, — (?a (where Ta = 1) the vector perpendicidar from 
the centre on the axis i, and let the vector 

be perpendicular to the plane of the circle. 

The equations of the circle are 

(p— tfa)* -h i« + c« = 0, 1 

S(i + I ia) p = 0. J 

Also — p« = 8Hp + S^a(i -^-SKiap, 

= SHp^8^ap'{-—SHp 

by the second of the equations of the circle. But, by the first, 

which is easily transformed into 

{p^^b^y =z^Aa^{p^ + 8Hp), 
or p'-^b* zzz^2aTFip. 

If we put this in the forms 

p»-i« = 2a8pp, 
and (p— flf)3)» + c* = 0, 

where ^ is a unit- vector perpendicular to i and in the plane of 
i and p, we see at once that the surface will be traced out by 
a circle of radius c, revolving about i, an axis in its own plane, 
distant a from its centre. 

This problem is not well adapted to show the gain in brevity 
and distinctness which generally follows the use of quaternions ; 


as, from its very nature, it hints at the adoption of rectangular 
axes and scalar equations for its treatment, so that the solution 
we have given is but little different from an ordinary Cartesian 

324. A surface is geiieraied ly a straight line which intersects 
two fixed Unes : fimd the genial equation. 

If the given lines intersect, there is no surface but the plane 
containing them. 

Let then their equations be. 

Hence every point of the surface satisfies the condition, § 30, 

9 = 5^(a+a?i8)4-(l-J^)(a,+^ii3i) (1) 

Obviously y may have any value whatever: but to specify a 
psurticular surface we must have a relation between x and x^. 
By tbe help of this, x^ may be eliminated from (1), which then 

takes the usual form 


Or we may operate on (1) by r'.(a+ir/3--ai— iFijSi), so that 

we get a vector equation equivalent to two scalar equations 

(§ 98), and not containing y. From this x and a?i may easily 

be found in terms of p, and the general equation of the possible 

surfeces may be written 

f{Xy X^ = 0, 

where/ is an arbitrary scalar function, and the values of x and 
Xy are expressed in terms of p. 

This process is obviously applicable if we have, instead of two 
straight lines, any two given curves through which the line 
must pass; and even when the tracing line is itself a given 
curve, situated in a given manner. But an example or two will 
make the whole process clear. 

325. Suppose the moveable line to be restricted by the condition 
that it is always parallel to afi^edplane. 

230 QUATERNIONS. [chjs:p. IXt 

Then, in addition to (1)^ we have the condition 

5y(at+«ti3x-o-ar)8) = 0, 

y being a vector perpendicular to the fixed plane. 

We lose no generality by asBiiming a and a^, which are any 
vectors drawn from the origin to the fixed lines^ to be each 
perpendicular to y; for, if for instance we could not assume 
Sya = 0, it would follow that SyP = 0, and the required surfiace 
would either be impossible, or would be a plane, cases which we 
need not consider. Hence 

^iSyfif^Syp = 0. 

Eliminating t^, by the help of this equation, from (1) of last 
section, we have 

Operating by any three non-coplanar vectors, we obtain three 
equations from which to eliminate x and y. Operating by S.y 
we find Syp = xSfiy. 

Eliminating x by means of this, we have finally 

which appears to be of the third order. It is really, however, 
only of the second order, since, consistently with our assump- 
tions, we have jr^^ y ^^ 

and therefore Syp is a spurious factor of the left-hand side. 

326. IJet the fixed Ivaee be perpendicular to each other, and 
let the moveable line pass through the drcumferenee of a circle, 
whose centre ie in the common perpendicular, and whose plane 
bisects that line at right angles. 

Here the equations of the fixed lines may be written 

p = a-\-xp, p =— oH-a?i.y, 


where a, fi, y form a rectangular system^ and we may assume 
the two latter to be unit- vectors. 
The circle has the equations 

p« =— «% Sap = 0. 
Equation (1) of § 324 becomes 

P=y(a + a?^) + (l— y)(-a-|-«?iy). 
Hence Sa-'^p = ^—(1 — ^) = 0, or ^ = h 

Also p= =— a» = (2y-l)«a» — ;ry-ar?(l— ^)S 

or 4 a' = (iJ?'-f-^i), ' 

so that if we now suppose the tensors of fi and y to be each 2 a, 
we may put a? = cos d, x^ = sin 6, from which 

p = (2y— l)aH-^/3cos^+(l— ^)ysin^; 
and finally -j- — P^^-rr + ,:. — o'^r^:, = 4a*. 

For this very simple case the solution is not better than the 
ordinary Cartesian one; but the student will easily see that 
we may by very slight changes adapt the above to data far less 
symmetrical than those from which we started. Suppose, for 
instance, p and y not to be at right angles to one another; 
and suppose the plane of the circle not to be parallel to their 
plane, &c., &c. But farther, operate on every line in space by 
the linear and vector function <^, and we distort the circle into 
an ellipse, the straight lines remaining straight. If we choose 
a form of <^ whose principal axes are parallel to a, p, y, the data 
wiU remain symmetrical, but not unless. This subject will be 
considered again in the next Chapter. 

327. To find the curvature of a normal section of a central 
surface of the second order. 

In this, and the few similar investigations which foUow, it 
will be simpler to employ infinitesimals than difi*erentials ; though 

232 QUATEBNIONS. [chap. IX- 

for a thorough treatment of the subject the latter method^ bb 

maj be seen in Hamilton's ElemenU, is preferable. 

We have, of coursej 

Spi^p = 1, 

and, if p+Sp be also a vector of the surface, we have rigorously, 
whatever be the tensor of bp, 

5(P+»P)*(P+»P) = 1- 

Hence 2Sbp<l>p-{-Shpipbp = (1) 

Now ^p is normal to the tangent plane at the extremity of p, 
so that if t denote the distance of the point p + bp from that 
plane ^ ^ SbpU<l>p, 

and (1) may therefore be written 

2tHl>p^T*bp8.Uip<l>mp = 0. 
But the curvature of the section is evidently 

or, by the last equation. 

In the limit, dp is a vector in the tangent plane ; let cr be the 
vector semidiameter of the surface which is parallel to it, and 
the equation of the sm&ce gives 

T^^S.U^U'SF = I, 

so that the curvature of the normal section, at the point p, in 

the direction of cr, is 


directly as the perpendicular from the centre on the tangent plane, 
and inversely as the square of the semidiameter parallel to the 
tangent line, a well-known theorem. 

.328. By the help of the known properties of the central 


section parallel to the tangent plane^ this theorem gives us all the 
ordinary properties of the directions of maximum and minimum 
curvature, their being at right angles to each other, the curva- 
ture in any normal section in terms of the chief curvatures and 
the inclination to their planes, &c., fee, without farther analysis. 
And when, in a future section, we show how to find an osculating 
surface of the second order at any point of a given surface, the 
same properties will be at once established for surfaces in 
general. Meanwhile we may prove another curious property 
of the surfaces of the second order, which similar reasoning 
extends to all surfaces. 

The equation of the normal at the point p+5p in the surface 
treated in last section is 

isr = p'\-lp + x<t>{p-\-hp) (1) 

This intersects the normal at p if (§§ 203, 210) 

S.bp(l>p(l)bp = 0, 

that is, by the result of § 273, if 6p be parallel to the maximum 
or minimum diameter of the central section parallel to the 
tangent plane. 

Let (Ti and o-, be those diameters, then we may write in 
&^^e^*al bp =pa;-^qir„ 

where jD and q are scalars, infinitely small. 

If we draw through a point-P in the normal at p a line 
parallel to o-i, we may write its equation 

-BT = p-\'(l<l>p+ya'i. 

The proximate normal (1) passes this line at a distance (see 

§ 2^^) 8.{a4>p^hp)UV(T^^{p-^hp), 

or, neglecting terms of the second order, 

•■^p^—{ap8.<l>pa^<l>a^-]-aqS.(l)par^<t>ar^ + qS,a^(ri(l>p). 

The first term in the bracket vanishes because a^ is a principal 



▼ector of the section parallel to the tangent plane^ and thus the 
expression becomes 

Hence^ if we take a s T<r\y the distance of the normal from the 
new line is of the second order only. This makes the distance 
of P from the point of contact T^^p Ta\y i. e. the principal radius 
of curvature along the tangent line parallel to <r,. That is the 
group of normals drawn near a point of a central surface of the 
second order pass ultimately throttgh two lines each parallel to the 
tangent to mie principal section, and passing through the centre of 
curvature of the other. The student may form a notion of the 
nature of this proposition by considering a small square plate^ 
with normals drawn at every point, to be slightly bent, but 
by different amounts, in planes perpendicular to its edges. The 
first bending will make all the normals pass through the axis 
of the cylinder of which the plate now forms part ; the second 
bending will not sensibly disturb this arrangement, except by 
lengthening or shortening the line in which the normals meet, 
but it will make them meet also in the axis of the new cylinder, 
at right angles to the first. A small pencil of light, with its 
focal lines, presents a similar appearance. 

329. To extend these theorems to surfaces in general, it is 
only necessary, as Hamilton ^has shown, to prove that if we 

^*« dv = i>dp, 

<^ is a self-conjugate function ; and then the properties of <f>, as 
explained in preceding Chapters, are applicable to the question. 

As the reader will easily see, this is merely another form 
of the investigation contained in § 317. But it is given here 
to show what a number of very simple demonstrations may be 
given of almost all quaternion theorems. 

Now V is defined by an equation of the form 
dfp = Svdp, 


where / is b, scalar fanction. Operating on this by another 
independent symbol of differentiation, b, we have 

bdfp = Sbvdp+Svhdp* 
In the same way we have 

db/p = Sdvbp+Svdbp, 
But, as d and b are independent, the left-hand members of these 
equations, as well as the second terms on the right (if these 
exist at all), are equal, so that we have 

Sdvbp = Sbvdp, 
This becomes, putting dv = Kpdp, 
and therefore bv = <t>bp, 

Sbp<f>dp = Sdp<l>bp, 
which proves the proposition. 

330. If we write the differential of the equation of a surface 
in the form dfp^2Svdp, 

then it is easy to see that 

/{p + dp) =z/p-\-2Svdp + 8dvdp-\-&c., 

the remaining terms containing as factors the third and higher 
powers of Tdp, To the second order, then, we may write, except 
for certain singular points, 

= 2Svdp-\-8dvdp, 
and, as before, (§ 327), the curvature of the normal section 
whose tangent line is dp is 


SSL The step taken in last section, although a veiy simple 
one, virtually implies that the first three terms of the expan- 
sion o{ f{p-\-dp) are to be formed in accordance with Taylor^s 
Theorem, whose applicability to the expansion of scalar functions 
of quaternions has not been proved in this work, (see § 135); 

H h !% 


we therefore give another investigation of the cu^rvature of a 
normal section^ employing for that purpose the formulae of 
§ (282). 
We have^ treating dp as an element of a curve^ 

Svdp = 0, 
or^ making 9 the independent variable, 

8vp' = 0. 
From this, by a second differentiation, 

S^p'+Si^p'' = 0. 

The curvature is, therefore, since v^p' and Tp = 1, 

332. Since we have shown that 
dv = ffidp 
where ^ is a self-conjugate linear and vector function, whose 
constants depend only upon the nature of the surface, and the 
position of the point of contact of the tangent plane ; so long 
as we do not alter these we must consider ^ as possessing the 
properties explained in Chapter V. 

Hence, as the expression for Tp" does not involve the tensor 

of dpy we may put for dp bxlj unit- vector r, subject of course to 

the condition 

8vT = (1) 

And the curvature of the normal section whose tangent is r is 

If we consider the central section of the surface of the second 

made by the plane Svzr = 0, 

we see at once that the curvature of ike gwen surface along the 


normal section touched by r is inversely as the square of the parallel 
radius in the auxiliary surface. This, of course, includes Euler^s 
and other well-known Theorems. 

333. To find the directions of maximum and minimum curvature^ 

we have 

/Sr<^T = max. or min. 

with the conditions, 

8VT = 0, 

?V = 1. 

By differentiation, as in § 273, we obtain the farther equation 
8.VTil>T = (1) 

If T be one of the two required directions rz=.rJJv is the other, 
for the last-written equation may be put in the form 

8.tUv<I){vtUv) = 0, 

i.e. 8.T<l^(v/) = 0, 

or S,vT'(f)T' = 0. 

Hence the sections of greatest and hast curvature are perpendicular 
to one another. 

We easily obtain, as in § 273, the following equation 

8.v{<l> + 8T(t>T)'^v = 0, 

whose roots divided by Tv are the required curvatures. 

334. Before leaving this very brief introduction to a subject, 
an exhaustive treatment of which will be found in Hamilton's 
Elements y we may make a remark on equation (1) of last section 

8.vt4)t = 0, 

or, as it may be written, by returning to the notation of § 332, 

S.vdpdv = 0. 

This is the general equation of lines of curvature. For, if we 
define a line of curvature on any surface as a line such that 

238 QUATERNIONS. [chap. IX 

nonnals drawn at contigaous points in it intersect^ then, bp 
being an element of such a line, the normals 

fir = p-^xp and fir = p-^bp-\-y{v^bv) 

must intersect. This gives, by § 203, the condition 

S.bpvhi; = 0, 
as above. 


1. Find the length of any arc of a curve drawn on a sphere 
so as to make a constant angle with a fixed diameter. 

2. Show that, if the normal plane of a curve always contains 
a fixed line, the curve is a circle. 

3. Find the radius of spherical curvature of the curve 

p = 4>t. 

Also find the equation of the locus of the centre of spherical 

4. (Hamilton, Bishop Lwufa "Premium Examination^ 1854.) 

{a,) If p be the variable vector of a curve in space, and if 
the differential dK be treated as = 0, then the 
equation if^_^^ ^ o 

obliges ic to be the vector of some point in the 
normal plane to the curve. 


(4.) In like manner the system of two equations, where 
dK and d*K are each = 0, 

dT{p^K) = 0, J>y(p-ic) = 0, 
represents the axis of the element, or the right line 
drawn through the centre of the osculating circle, 
perpendicular to the osculating plane. 

(c.) The system of the three equations, in which k is 
^ treated as constant, 

determines the vector k of the centre of the oscu- 
lating sphere. 

(d.) For the three last equations we may substitute the 

8\p'-K)dp = 0, 

S.(p-^K)d^p-^3S.dpd'p = 0. 

(e.) Hence, generally, whatever the independent and scalar 
variable may be, on which the variable vector p of 
the curve depends, the vector k of the centre of the 
osculating sphere admits of being thus expressed : 
_ ^V.dpd^pS.dpd^p-dp^F.dpd^p 
''-P-^ S.dpd^pd^p 

(/.) In general, 

did-^r.dpUp) = d(Tp-*r.pdp) 

= Tp-'{3F.pdp8.pdp^p'r.pd'p); 

sr.pdpS.pdp-^p^F.pd'p = 'p^Tpdip-^r.dpUp); 
and, therefore, the recent expression for k admits of 
being thus transformed, 

_ dp*d{dp-^F. d^pUdp) 
"-P-^ S.d'pd'pUdp ' 

{ff,) If the length of the element of the curve be constant, 
dTdp = 0, this last expression for the vector of the 

240 QUATBBNIONS. [chap. IX. 

centre of the osculating^ sphere to a curve of doahle 
curvature becomes^ more simply, , 

"" ^ ^'^ S.dpd^pd^p' 

or r-nJ. ^'^'P^P' 


(i.) Verify that this expression gives k = 0, for a curve 
described on a sphere which has its centre at the 
origin of vectors ; or show that whenever dTp = 0, 
d^Tp = 0, d^Tp = 0, as well as dTdp = 0, then 

pS.dp'^d^pd'p = V.dpd^p. 

6. Find the curve from every point of which three given 
spheres will appear of equal magnitude. 

6. Show that the locus of a point, the difference of whose 
distances from each two of three given points is constant, is 
a plane curve. 

7. Find the equation of the curve which cuts at a given angle 
all the sides of a cone of the second order. 

Find the length of any arc in terms of the distances of its 
extremities from the vertex. 

8. Why is the centre of spherical curvature, of a curve de- 
scribed on a sphere, not necessarily the centre of the sphere ? 

9. Find the equation of the developable surface whose gene- 
rating lines are the intersections of successive normal planes to a 
given tortuous curve. 

10. Find the length of an arc of a tortuous curve whose 
normal planes are equidistant from the origin. 

11. The reciprocals of the perpendiculars from the origin on 
the tangent planes to a developable surface are vectors of a 
tortuous curve; from whose osculating planes the cusp-edge 
of the original surface may be reproduced by the same process. 


12. The equation 

where a is a unit- vector not perpendicular to j3, represents an 
ellipse. If we put y = Fa/3, show that the equations of the 
locus of the centre of curvatui*e are 

S.fiyp = 0, 
S^fip + S^yp = (fiSUafi)i. 

13. Rnd the radius of absolute curvature of a spherical conic. 

14. If a cone be cut in a circle by a plane perpendicular to a 
side^ the axis of the right cone which osculates it^ along that 
side, passes through the centre of the sectioiv. 

15. Show how to find the vector of an umbilicus. Apply 
your method to the surfaces whose equations are 

aS/m^P = 1, 

and SapSppSyp = 1. 

16. Find the locus of the umbilici of the surfaces represented 
by the equation 

8p{<t> + Ay'p = 1, 

where ^ is an arbitrary parameter. 

17. Show how to find the equation of a tangent plane which 
touches a surface along a line. Find such planes for the follow- 
ing surfaces 

Sp<l>p = 1, 

and (p^— a=»-f i»)» + 4(aV' + *'^'ap) = 0. 

18. Find the condition that the equation 


where is a self-conjugate linear and vector function, may 
represent a cone. 

I i 


19. Show from the general equation that cones and cylinders 
are the only developable surfaces of the second order. 

20. Find the equation of the envelop of planes drawn at each 
point of an ellipsoid perpendicular to the radius vector from the 

21. Find the equation of the envelop of spheres whose centres 
lie on a given sphere, and which pass through a given point. 

22. Find the locus of the foot of the perpendicular from the 
centre to the tangent plane of a hyperboloid of one, or of two, 

23. Hamilton, Bishop Lavfs Premium Examinationy 1852. 

(a.) If p be the vector of a curve in space, the length of 
the element of that curve is Tdp ; and the variation 
of the length of a finite arc of the curve is 
h/Tdp = -/iS TJdphdp = ^l^SUdphp +/SdUdphp. 

(b.) Hence, if the curve be a shortest line on a given sur- 
face, for which the normal vector is v, so that 
Svbp = 0, this shortest or geodetic curve must 
satisfy the differential equation, 
FpdUdp = 0. 
Also, for the extremities of the arc, we have the 
limiting equations, 

SUdp.bpo = ; SUdp^bpi = 0. 
Interpret these results. 

((?.) For a spheric surface, Fvp = 0, pdUdp = ; the inte- 
grated equation of the geodetics is pUdp = w, giving 
Surp = (g^eat circle). 
For an arbitrary cylindric surface, 

8av = 0, adUdp = ; 
the integral shows that the geodetic is generally a 
helix, making a constant angle with the generating 
lines of the cylinder. 


(d.) For an arbitrary conic surface, 

Svp = 0, SpdUdp = ; 

integrate this differential equation, so as to deduce 

from it, TFpUdp = const. 
Interpret this result; shew that the perpendicular 

from the vertex of the cone on the tangent to a 

given geodetic line is constant; this gives the 

rectilinear development. 
When the cone is of the second degree, the same 

property is a particular case of a theorem respecting 

confocal surfaces. 

{e,) For a surface of revolution, 

S.cLpv = 0, S.apdUdp = ; 

integration gives, 

const = S.apUdp = TFapSUiFapJp) ; 

the perpendicular distance of a point on a geodetic 
line from the axis of revolution varies inversely as 
the cosine of the angle under which the geodetic 
crosses a parallel (or circle) on the surface. 

(/.) The differential equation, S.apdUdp = 0, is satisfied not 
only by the geodetics, but also by the circles, on a 
surface of revolution ; give the explanation of this 
fact of calculation, and show that it arises from the 
coincidence between the normal plane to the circle 
and the plane of the meridian of the surface. 

(ff.) For any arbitrary surface, the equation of the geodetic 
may be thus transformed, S.vdpd^p — ; deduce 
this form, and show that it expresses the normal 
property of the osculating plane. 

(L) If the element of the geodetic be constant, dTdp = 0, 
then the general equation formerly assigned may be 
reduced to V.vd^p = 0. 
Under the same condition, d^p = —v-^Sdvdp. 
I i % 

244 QUATERNIONS. [chap. IX. 

(>.) If the equation of a central surface of the second order 
be put under the form ^=: 1, where the fanetion/ 
is scalar, and homogeneous of the second dimension^ 
then the differential of that function is of the form 
dfy = 2S.vdp, where the normal vector, r = <^p, is a 
distributive function of p (homogeneous of the first 
dimension), dv = d<l>p = <f>dp. 
This normal vector v may be called the vector of prox^ 
imity (namely, of the element of the surface to the 
centre); because its reciprocal, r~*, represents in 
length and in direction the perpendicular let fall 
from the centre on the tangent plane to the sur&ce. 

(^.) If we make Sa^p z=.f{iTy p), thi% function f is com- 
mutative with respect to the two vectors on which 
it depends, y(p, o) =/(a', p) ; it is also connected 
with the former function f of a single vector p, by 
the relation, y(p, p) =/p : so thatyp = &p<^p. 
fdp = Sdpdv ; dfdp = 2S.dvd'p ; for a geodetic, with 
constant element, 

2fdp V 

this equation is immediately integrable, and gives 
const. = Tv ^/{fUdp) = reciprocal of Joachimstal^s 
product, PL. 

(/.) If we give the name of " Didonia'' to the curve (dis- 
cussed by Delaunay) which, on a given surface and 
with a given perimeter, contains the greatest area, 
then for such a Didonian curve we have by quater- 
nions the formula, 

fS. Uvdphp -f chfTdp = 0, 
where c is an arbitrary constant. 
Derive hence the differential equation of the second 
order, equivalent (through the constant c) to one 
of the third order, 

c-'dp = VMvdUdp. 


Geodetics are, therefore, that limiting case of Didonias 

for which the constant c is infinite. 
On a plane, the Didonia is a circle, of which the 

equation, obtained by integration from the general 

form, is 

-or being vector of centre, and c being radius of 

{m,) Operating by 8. Udpy the general differential equation 
of the Didonia takes easily the following forms : 

c-^Tdp =S{Uvdp.dUdp); 

c-^Tdp^ =:8{Ui;dp.d'p); 

C-' Tdp' = 8. Uvdpd^p ; 

, d^pdp-^ 

= 8 


{n.) The vector ©, of the centre of the osculating circle to 
a curve in space, of which the element Tdp is con- 
stant, has for expression, 


« = '' + -^- 

Hence for the general Didonia, 


T{p--<^) = c8U^. 
^ ^ vdp 

{o,) Hence, the radius of curvature of any one Didonia 
varies, in general, proportionally to the cosine of the 
inclination of the osculating plane of the curve to 
the tangent plane of the surface. 
And hence, by Meusnier's theorem, the difference of 
the squares of the curvatures of curve and surface 
is constant; the curvature of the surface meaning 

246 QUATERNIONS. [chap. IX. 

here the reciprocal of the radius of the sphere which 
oscuhites in the redaction of the element of the 

(p.) In general, for any curve on any surface, if £ denote 
the vector of the intersection of the axis of the 
element (or the axis of the circle osculating to the 
curve) with the tangent plane to the surface, then 

^ " ''■^ S.vdpd'p ' 

Hence, for the general Didonia, with the same signi- 
fication of the symbols, 

$ = p—cUvdp; 
and the constant c expresses the length of the in- 
terval p— f, intercepted on the tangent phine, be- 
tween the point of the curve and the axis of the 
osculating circle. 

(q,) If, then, a sphere be described, which shall have its 
centre on the tangent plane, and shall contain the 
osculating circle, the radius of this sphere shall 
always be equal to e, 

(r.) The recent expression for (, combined with the first 
form of the general differential equation of the 
Didonia, gives 

d( = -cFdUvUdp ; Vvd^ = 0. 

(«.) Hence, or from the geometrical signification of the 
constant <?, the known property may be proved, 
that if a developable surface be circumscribed about 
the arbitrary surface, so as to touch it along a 
Didonia, and if this developable be then unfolded 
into a plane, the curve will at the same time be 
flattened (generally) into a circular arc, with radius 
= c. 

24. Find the condition that the equation 


may give three real values oi f for any given value of p. lif 
be a fiinction of a scalar parameter £, show how to find the form 
of this function in order that we may have 

„.> d'^ . d^e d'i ^ 
^~ dx' ^ dy ^ dz* ~ ' 

Prove that the following is the relation between^ and f, 
■rr-f ^^ ^{^^ 

in the notation of § 147. 

25. Show, after Hamilton, that the proof of Dupin's theorem, 
that ^^ each member of one of three series of orthogonal surfaces 
cuts each of the other series along its lines of curvature, '^ may 
be expressed in quaternion notation as follows : 

If Svdp = 0, Sv'dp = 0, 8,vv'dp^0 

be integrable, and if 

Svv=^ 0, then Vvdp = makes S.vvdif = 0. 

Or, as follows. 

If /S'i;Vi; = 0, ^/V2/ = 0, Sv'Vv"^^, and r.wV'= 0, 

then S,v'\8vV.v) = 0, 

1 ' T-, ' d , d , d 

where v = t-=- + 9 -t~ +^-7- • 

dx "^ dy dz 

26. Show that the equation 

Vap = pY^p 

represents the line of intersection of a cylinder and cone, of the 
second order, which have )3 as a common generating line. 



335. \\7 *^cn a point's vector, p, is a Ainction of the time 

▼ f ^, we have seen (§ 35) that its vector- velocity is 

expressed by --^ or, in Newton^s notation, by p. 

That is, if p = <t>t 

be the equation of an orbit, containing (as the reader may see) 
not merely the form of the orbit, but the law of its description 
also, then p = tl/t 

gives at once the form of the Hodograph and the law of its 

This shows immediately that the vector^acceleration (fapoinfs 
motion ^«p 

is the vector-velocity in the hodograph. Thus the fimdamental 
properties of the hodograph are proved almost intuitively. 

336. Changing the independent variable, we have 

dp ds , 

if we employ the dash, as before, to denote -j- • 

This merely shows, in another form, that p expresses the 
velocity in magnitude and direction. But a second differentia- 
tion gives p = vp'\-v^p\ 

This shows that the vector-acceleration can be resolved into two 
components, the first, vp, being in the direction of motion and 

SECT. 338.] KINEMATICS. .249 

equal in magnitade to the acceleration of the velocity^ v or -^', 

the second, v^p'y being in the direction of the radius of absolute 
curvature, and its amount equal to the square of the velocity 
multiplied by the curvature. 

[It is scarcely conceivable that this important fundamental 
proposition, of which no simple analytical proof seems to have 
been obtained by Cartesian methods, can be proved more ele- 
gantly than by the process just given,] 

337. If the motion be in a plane curve, we may write the 
equation as follows, so as to introduce the usual polar coor- 
dinates, T and 6, 20 

P = ^a'/3, 

where o is a imit vector perpendicular to, j8 a unit vector in, the 
plane of the curve. 

Here, of course, r and 6 wrj be considered as connected by 
one scalar equation,* or better, each may be looked on as a 
function of t. By differentiation we get 

which shows at once that r is the velocity along, rO that per- 
pendicular to, the radius vector. Again, 

3£ . . ,.2(9 

which gives, by inspection, the components of acceleration 
along, and perpendicular to, the radius vector. 

338. For imiform acceleration in a constant direction, we have 

at once, 

p =z a. 

Whence p = d^+ft 

where j8 is the vector-velocity at epoch. This shows that the 
hodogiaph is a straight line described uniformly. 



Also p = ~^pt, 

no constant being added if the origin be assumed to be the 
position of the moving point at epoch. 

Since the resolved parts of p, parallel to p and o, vary re- 
spectively as the first and second powers of t, the curve is evi- 
dently a parabola (§31 (/)). 

But we may easily deduce from the equation the following 

the equation of a paraboloid of revolution, whose axis is a. 
Also S.afip = 0, 

and therefore the distance of any point in the path from the 
point ~|)3a-^j3 is equal to its distance from the line whose 
equation is 

p = — ^ a"* +«a Fap. 
Thus we recognise the focus and directrix property. 

339. That the moving point may reach a point y we must 
have, for some real value of ty 

Now suppose T^y the velocity of projection, to be given = r, 
and, for shortness, write «r for {7)3. 

Then y = ^P^-vtm. 


Since Tta =1, , 

we have (t;* — 8ay) ^* -f ^' = 0. 

The values of P are Hal if 


8SCT. 341.] KINBHATICS. 251 

is positive. NoWj as TaTy is never less than Say^ it is evident 
that v^-'Say must always be positive if the roots are possible. 
Hence, when they are possible, both values of t* are .positive. 
Thus we have four values of t which satisfy the^ conditions, and 
it is easy to see that since, disregarding the signs, they are 
equal two and two, each pair refer to the same path, but de^ 
scribed in opposite directions between the origin and the extremity 
of y. There are therefore, if any, in general two parabolas 
which satisfy the conditions. The directions of projection kre 
(of course) given by the corresponding values of «r. 

340. The envelop of aU the trajectories possible with a given 
velocity, evidently corresponds to 

{v^^SayY-Ta^Ty^ = 0, 
for then y is the vector of intersection of two indefinitely close 
paths in the same vertical plane. 

Now v*^Say=zTaTy 

is evidently the equation of a paraboloid of revolution of which 
the origin is the focus, the axis parallel to a, and the directrix 

plane at a distance ^p • 

All the ordinary problems connected with parabolic motion 

are easily solved by means of the above formulae. Some, 

however, are even more easily treated by assuming a horizontal 

unit vector in the plane of motion, and expressing p in terms of 

it and a. But this must be left to the student. 

34L For acceleration directed to or from a fixed point, we have, 
taking that point as origin, and putting P for the magnitude of 
the central acceleration, 

p = PUp. 
Whence, at once, Vpp = 0. 

Integrating, Vpp = y == a constant vector. 

The interpretation of this simple formula v&^^fi^st, p and p are 

X k (z 


in a plane perpendicular to y, hence the path is in a plane (of 
course passing through the origin); second, the area of the 
triangle, two of whose sides are p and p, is constant. 

[It is scarcely possible to imagine that a more simple proof 
than this can be given of the fondamental &ct8, that a central 
orbit is a plane curve, and that equal areas are described by the 
radius vector in equal times.] 

842. When the law of acceleraium to or from tie origin is 
Hat of the inv^se square of tie distance, we have 

where fx is negaOoe if the acceleration be directed to the origin. 

Hence p = ^. 

The following beautiful method of integration is due to 
Hamilton. (See Chapter lY.) 

a^a^Wrr ^^^ — Up .Vpp _ Up.y 



and py =:€— fiCTp, 

where € is a constant vector, perpendicular to y, because 

Syp = 0. 
Hence, in this case, we have for the hodograph, 
p = ey-^—nUp.y-^. 

Of the two parts of this expression, which are both vectors, 
the first is constant, and the second is constant in length. 
Hence the locus of the extremity of p is a circle, whose radius 

is ^, and whose centre is at the extremity of the vector ey-K 

^ [This equation contains the whole theory of the Circular 

8BCT. S44.] KINEMATICS- 253 

Hodoffraph. Its consequences are developed at length in Hamil- 
ton's ElemenU."] 

343. We may write the equations of this circle in the form 

(a sphere), and 8yp = 0, 

(a plane through the origin^ and through the centre of the 


The equation of the orbit is found by operating by V.p upon 
that to the hodograph. We thus obtain 

or y« = 8€p-\-yLTp, 

or iyi2}) = ^€(y^€-»-p); 

in which last form we at once recognise the focus and directrix 
property. This is in fact the equation of a conicoid of revolu- 
tion about its principal axis {€)y and the origin is one of the 
foci. The orbit is found by combining it with the equation of 
its plane, Syp = 0. 

We see at once that y*€-* is the vector distance of the directrix 

from the focus; and similarly that the excentricity is — , and 
the maior axis , , • 

344. To take a simpler case : let the acceleration vary as the 
dUtancefram the orujfin. 

Then p = ±wV, 

the upper or lower sign being used according as the acceleration 
iajrom or to the centre. 

(d* \ 

-^^:m^)p = 0. 

Hence . p= as'^'+jSr-"*'; 


or p= aoosmt-^-fimnrnt, 

where a and p are arbitrary, but constant, vectors ; and € is the 
base of Napier's logarithms. 

The first is the equation of a hyperbola of which a and fi are 
the directions of the asymptotes ; the second, that of an ellipse 
of which a and fi are semi-conjugate diameters. 

Since p = «» {ctf^— j8£~"^}, 

or = «{— asin«i^-|-)3cos«»^}, 

the hodograph is again a hyperbola or ellipse. But in the first 
case it is, if we neglect the change of dimensions indicated by 
the scalar factor m, conjugate to the orbit ; in the case of the 
ellipse it is similar and similarly situated. 

346* Again, let the aeceleratian be as the inverse third power 
ofihe distance, we have 

Of course, we have, as usual, 

Fpp = y. 
Also, operating by S.p, 


Spp = 

of which the integral is 

the equation of energy. 

Tp' * 

p.= C7-A 

Again, Spp=zJ^. 


Hence Spp+p* = C, 

or Spp =1 Ct, 

no constant being added if we reckon the time from the passage 
through the apse, where Spp = 0. 

We have, therefore, by a second integration, 

p« = Ct* + Cr. ] (1) 


[To determine Cy remark that 

or p^p^ =zCH^-^y\ 

But p«p« = Cp^^fjL (by the equation of energy), 

= CH^ + C(r^,M, by(l). 
Hence, (7C = m— y'O 

To complete the solution, we have, by § 133, 

where )S is a unit vector in the plane of the orbit. 

But ri=_2.. 

P P' 

Hence log-^=_y/^. 

The elimination of t between this equation and (1) gives Tp 
in terms of Up, or the required equation of the path. 

We may remark that if d be the ordinary polar angle in the 

^''^'*' I ^P nrr 

log-j- = euy. 
Hence we have 

and r*=i^{Ct* + C% 

from which the ordinary equations of Cotes^ spirals can be at 

once foimd. 

346. To find tie conditions that a given curve may be the 
hodogra^h corresponding to a central orbit. 

If «r be its vector, gi/oen as a function of the time, fvr dt is 
that of the orbit; hence the requisite conditions are given by 

Yrafrndt = y, 
where y is a constant vector. 


We may transform this into other shapes more resembling 
the Cartesian ones. 

Thus Vmfmdi = 0, 

and Y^fiadi'\- F«rw = 0. 

From the first fmdi = X'usj 

and therefore x Fmr = y, 

or the carve is plane. And 

X Vvir + Virv = ; 
or eliminating a?, y Vmr = — (FW«r)*. 
Now if t;' be the velocity in the hodograph, K its radius of 
curvature^ j/ the perpendicular on the tangent; this equation 
gives at once hif ^ Rp'\ 

which agrees with known results. 

347. The equation of an epitrocAaid or AypotrocAoid, referred 
to the centre of the fixed circle^ is evidently 

p ^ ai* a'{-bi' a^ 
where a is a unit- vector in the phme of the curve and i another 
perpendicular to it. Here a> and 6>| are the angular velocities 
in the two circles^ and t is the time elapsed since the tracing 
point and the centres of the two circles were in one straight 

Hence, for the length of an arc of such a curve, 

# =/Tprf^ =/£»>/{a)»tf«H-2fiM»iaAcos(»— «i)^+«i'**h 

which is, of course, an elliptic Amotion. 

But when the curve becomes an epicycloid or a hypocycloid^ 
a)a+»id s= 0, and 

SECT. 348.] KINEMATICS. 257 

which can be expressed in finite terms, as was first shown by 
Newton in the Prindpia, 

The hodograph is another curve of the same class, whose 
equation is 2»t 2uit 

p = i(a<ai ' o-f^ia)ii ^ a) j 

land the acceleration is denoted in magnitude and direction by 
the vector 2«e 2^ 

p = — aa)*i ^ a-'hfali » a. 

Of course the equations of the common Cycloid and Trochoid 
may be easily deduced from these forms by making a indefinitely 
great and a> indefinitely small, but the product ao) finite ; and 
transferring the origin to the point 

p = aa. 

348. Let i be the normal- vector to any plane. 
Let 'CT and p be the vectors of any two points in a rigid plate 
in contact with the plane. 

After any small displacement of the rigid plate in its plane, 
let dvT and dp be the increments of tsr and p. 

Then Sidrsr = 0, Sidp = ; and, since jr('Bj— p) is constant, 
iS(«r— p)(rf«T — dp) = 0. 
And we may evidently assume 

dp = <i>i(p--r), 
disT = a)i('CJ — t); 

where of course t is the vector of some point in the plane, to a 
rotation a> about which the displacement is therefore equivalent. 
Eliminating it, we have 


which gives w, and thence r is at once found. 
For any other point a in the plane figure 
Sida- = 0, 
S(j)^(r){dp'^d<T) = 0. Hence dp— da = u)ii(p — a). 

L 1 


5(<r— w)(Jttr— Ar) = 0. Henoe itr^dm = a>,i(<r— «r). 
From which^ at once^ «#, =r a>, = m, and 

dir = «t(<r— r), 
or this point also is displaced by a rotation a> about an axis 
through the extremity of r and parallel to u 

349. In the case of a rigid body moving about a fixed point 
let «r, py a denote the vectors of any three points of the body; 
the fixed point being origin. 

Then vt^y p*, o-* are constant, and so are Svrp, Spc, and 8(nr. 
After any small displacement we have, for ^ and p, 
Smdv = 0, " 

Spdp^O, (1) 

8mdp'\'Spdm =s 0. 
Now these three equations are satisfied by 
dor = VwuTj dp = Vap, 
where a is any vector whatever. But if d^ and dp are gtBen^ 
then Vdwdp = KFamVap = aS.apvr. 

Operate by S.Fisrp, and remember (1), 

S^mdp = S^pdm = S^.apw. 

Hence o = -^ = -^ (2) 

Now consider o-, Sadc = 0, 

Spda- ^—Sadp^ 
8md<T = — 8ader» . 
d<r = Fcur satisfies them all, by (2), and we have thus the 
proposition that any small displacement of a rigid body about a 
fixed point is equivalent to a rotation, 

350. To represent the rotation of a rigid body about a given 
axis, through a given finite angle. 

Let a be a unit-vector in the direction of the axis, p the 

«SCT. 351.] KINEMATICS. 259 

vector of any point in the body with reference to a fixed point 
in the axis^ and the angle of rotation. 

Then p = a-^ Sap + a~^ Fap, 

= — aSap — a Fap. 

The rotation leaves^ of courscj the first part unafiected^ but 
the second evidently becomes 

or --aFapQOsd'\-Fap8mO. 

Hence p becomes 

/>! = -^aSap — aFap cos 6 + Vap sin 6, 

= (cos- H-asin-)f)(cos- -asin-), 

35L Hence to compound two rotations about axes which meet, 
we may evidently write, as the effect of an additional rotation ^ 
about ibe unit- vector fi, 

Hence p, = p' a' pa 'fi '. 

If the j3-rotation had been first, and then the a-rotation, we 

should have had ^ t ± -* ^I 

p\ = a'p'^pp 'a ', 

and the non-commutative property of quaternion multiplication 

shows that we have not, in general, 

p\ = Pj. 
If «j A y be radii of the umt sphere to the corners of a 

spherical triangle whose angles are -, ^, ^, we know that 

A A M 

t ± I 

y^fi^'a' = — 1. (Hamilton, Lectures, p. 267.) 

L 1 !2 

260 QUATERNIONS. [chap. X. 

Hence /3'a'=-y"', 

_* t 
and we may write P* ^ y 'py' j 

OT, successive rotations about radii to two comers of a spherical 
triangle, and through angles double of those of the triangle, are 
equivalent to a single rotation about the radius to the third comer, 
and through an angle double of the exterior angle of the triangle. 

Thus any number of successive finite rotations may be com- 
pounded into a single rotation about a definite axis. 

352. When the rotations are indefinitely small^ the effect of 
one is, by § 350, ^^ «, p^eVap, 

and for the two, neglecting products of small quantities, 

p. = p-^OFap-i-ttiFfip, 
$ and if> representing the angles of rotation about the unit 
vectors a and /3 respectively. 
But this is equivalent to 

representing a rotation through an angle T{6a + <f>p), about the 
unit- vector U($a'h4>P)' Now the latter is the direction, and 
the former the length, of the diagonal of the parallelogram 
whose sides are 6a and <^/3. 

We may write these results more simply, by putting a for 
$a, p for if>p, where a and p are now no longer unit- vectors, but 
represent by their versors the axes, and by their tensors the 
angles (small), of rotation. 

Thus pi = p-f Fapy 

p, = p-j-Fap+Fpp, 

353. The general theorem, of which a few preceding sections 
illustrate special cases, is this : 

By a rotation, about the axis of q, through double the angle 

SECT. 354.] KINEMATICS. 261 

of q, the quaternion r becomes the quaternion qrq-^. Its tensor 
and angle remain unchanged^ its plane or axis alone varies. 

A glance at the figure is suffi- 

• Q cient for the proof, if we note 

that of course T.grq;-^ = Tr, and 

therefore that we need consider 

the versor parts only. Let Q be 

r/K^'^ ^^^ pole o^ 9> 

Join CB. 

Then CB is j'r^'-^, its arc CB is evidently equal in length to 
that of r, JS'C"; and its plane (making the same angle with 
B'B that that of JS'C" does) has evidently been made to revolve 
about Q, the pole of q, through double the angle of q. 

If r be a vector, = p, then qpq~^ is the result of a rotation 
through double the angle of q about the axis of q. Hence, as 
Hamilton has expressed it, if B represent a rigid system, or 
assemblage of vectors, 9.^97^ 

is its new position after rotating through double the angle oi q 
about the axis of q. 

354L To compound such rotations, we have 
r,qBq-^,r-^ = rq.B,{rq)-^, 
To cause rotation through an angle ^-fold the double of the 
angle of q we write i^Bq"^. 

To reverse the direction of this rotation write 

To translate the body B without rotation, each point of it 
moving through the vector a, we write 

To produce rotation of the translated body about the same 
axis, and through the same angle, as before, 

q{a + B)r'> 

262 QCATBRNIONS. [chap. X. 

Had we rotated first, and then translated, we should have had 

Tlie discrepance between these last results might perhaps be 
useful to those who do not believe in the Moon's rotation, but 
to such men quaternions are unintelligible. 

855* By the definition of Homogeneous Strain, it is evident 
that if we take any three (non-coplanar) unit- vectors a, ^3, y in 
an unstrained mass, they become after the strain other vectors, 
not necessarily unit- vectors, a^, )9i, y^. 

Hence any other given vector, which of course may be thus 

expressed, p = xa-^yfi-^zy, 

becomes pi = xa^ ^ypi + zy^ , 

and is therefore known if Oi, J8i, y^ be given. 

More precisely 

pS.apy = aS.pyp^-pS.yap^-yS.aPp 

Pi8.aPy = 4^p8.aPy == a^8.fiyp'\-fi^8.yap-\-yiS.aPp. 

Thus the properties of <^, as in Chapter V, enable us to study 
with great simplicity strains or displacements in a solid or 

For instance, to find a vector whose direction is unchanged by 
the strain, is to solve the equation 

Fpi^p = 0, or 4»p = gp, 
where ^ is a scalar unknown. 

[This vector equation is equivalent to three simple equations, 
and contains only three unknown quantities; viz. two for the 
direction of p (the tensor does not enter, or, rather, is a &ctor of 
each side), and the unknown gJ] 

We have seen that every such equation leads to a cubic in g 
which may be written 

g^—m^g^-\-myg^m = 0, 
where Wj, «»i, ;» are scalars depending in a known manner on 

SECT. 357.] 



the constant vectors involved in <f>. This must have one real 
root, and may have tkree. 

356, For simplicity let us assume that a, fi, y form a rect- 
angular system, then we may operate by 8,ay S.p, and S.y; and 
thus at once obtain the equation for ^, in the form 

Saa^-i-ff, Sap^, Say^ == (1) 

spa,, Spp.-^-g, spy, 

Sya,, Syp,, Syy.^g 

If the mass be rigid we must have Oj, ^i, y^ still rectangular 
unit- vectors, and therefore 

SaPt = Spa^,^ 

Say, = Sya,, (2) 

Syp, = SPy,,^ 

in which case (1) has obviously -fl as onfj root: the others 
being imaginary, except in two limiting cases in which their 
values are equal, and each is -|- 1 or — 1. 

[One simple method of obtaining the conditions (2) is to 
suppose q the quaternion by a rotation about whose axis and 
through double of its angle a, p, y are converted into Oi , yS, , yi . 
We have thus, by § 353, 

Oi = qaq-^ Sec, 
and equations (2) express the identities 

S.aqPgr^ = S.pqof^ &c. 
Numerous equally simple quaternion proofs will be obvious to 
the intelligent student.] 

357. If we take Tp =z C we consider a portion of the mass 
initially spherical. This becomes of course 

an ellipsoid, in the strained state of the body. 

Or if we consider a portion which is spherical after the strain, 

264 QIJATEKNIONS. [ghap. X. 

its initial form was T^p =s C, 

another ellipsoid. The relation between these ellipsoids is ob- 
vious from their equations. (See 311.) 

In either case the axes of the ellipsoid correspond to a rect- 
angular set of three diameters of the sphere (§ 257). But we 
must carefully separate the cases in which these corresponding 
lines in the two surfaces are^ and are not^ coincident. For^ in 
the former case there is pure strain, in the latter the strain is 
accompanied by rotation. Here we have at once the distinction 
pointed out by Stokes * and Helmholtz f between the cases of 
fluid motion in which there is, or is not, a velocity-potential. 
In ordinary fluid motion the distortion is of the nature of a 
pure strain, i. e. is differentially non-rotational ; while in vortex 
motion it is essentially accompanied by rotation. But the re- 
sultant of two pure strains is generally a strain accompanied 
by rotation. The question before us beautifiilly illustrates the 
properties of the quaternion linear and vector Amotion. 

358. To find the quaternion Jbrmnla for a pure strain. Take 
a, /3, y now as unit-vectors parallel to the axes of the strain- 
ellipsoid, they become aa, ij3, cy. 

Hence p^ = <l>p = —aaSap^bpSpp—cySyp, 
And we have, for the criterion of a pure strain, the property 
of the function <f», that it is self -conjugate y i. e. 

8p<f><T ^ 8<T<^p. 

359. Two pure strains, in succession, generally give a strain 
accompanied by rotation. For if <f», i^ represent the strains, since 
they are pure we have 

SpiKT = Safpp,^ 
8py^<T = Sinj/p. J 

* Cambridge PKU. Trans, 1845. 

t CrdU, vol. Iv, 1857. S^ iklso Phil. Mag. (Supplement) June 1867. 

SECT. 362.] KINEMATICS. 265 

But for the compound strain we have 

Pi=XP = ^4>pj 
and we have not generally 

Spxo- = Saxp* 
For Sp\j/<t>(r = Sa<t>^p, 

^y (1)> ^^^ V^<^ is not generally the same as (f>\lf. (See Ex. 7 
to Chapter V.) 

360. The simplicity of this view of the question might lead us 
to suppose that we may easily separate the pure strain from the 
rotation in any case, and exhibit the corresponding functions. 
But, for this purpose, it is generally necessary to solve the cubic 
equation of Chapter V in each particular case. When this is 
effected, the rest of the process presents no difficulty. 

361. In general, if 

Pi = <^P = — aj/Sap— ^j/S/Sp— yi/Syp, 
the angle between any two lines, say p and o-, becomes in the 
altered state of the body 

cos- '{—8.U(l>pU<t>(T). 
The plane SCp = becomes (with the notation of § 144) 
/S^p, = = 8C(t>p = 8p(l/C. 

Hence the angle between the planes 8(p = 0, and Srjp = 0, which 
is cos-^ {—SMCUrj), becomes 

cos-' {-S.U(t>W(p'ri)' 
The locus of lines equally elongated is, of course, 
Ti^Vp = e, 
or Ti^p = eTpy 

a cone of the second order. 

362. In the case of a Simple Shear y we have, obviously, 

Pi = (t>p = p-^pSap. 
M m 

266 QUATERNIONS. [chap. X. 

The yectors which are unaltered in length are g^ven by 

Tp, = Tp, 

or 2SfipSap + P*S*ap = 0, 

which breaks up into 

S.ap = 0, 

and Sp(2/3+/3«a) = 0. 

The intersection of this plane with the plane of a, fi is per- 
pendicular to 2/3+i3'a. Let it be a-{-xp, then 

5.(2/3-}-/3*a)(a+ar/3) = 0, 

i. e. 2ar— 1 = 0. 

Hence the intersection required is • 

For the axes of the strain^ one is of course afi, and the others 
are found by making T(l>Up 2l maximum and minimum. 


p = a+xp, 


Pi = *f> = a+ar/3--/3, 



-~- — max. or mm.. 



from which the values of x are found. 
Also, as a verification, 

S.(a4-^i/3)(a4-3r,j3) = -1 4-/3«^iar,, 
and should be = 0. It is so, since, by the equation. 

^{a4-(^i-l)^}{a4-(^,-l)/3}=-l+^»{^i^«-(^i+^,) + l}, 
which ought also to be zero. And, in fact, ^Tj+ara = 1 by the 
equation ; so that this also is verified. 

SECT. 364.] KINEMATICS. 267 

363. We regret that our limits do not allow us to enter 
farther upon this very beautiful application. 

Bat it may be interesting here^ especially for the consideration 
of any continuous displacements of the particles of a mass^ to 
introduce another of the extraordinary instruments of analysis 
which Hamilton has invented. Part of what is now to be 
g^ven has been anticipated in last Chapter, but for continuity 
we commence afresh. 

If Fp^C (1) 

be the equation of one of a system of surfaces, and if the differ- 
ential of (1) be 

Svdp=:0, (2) 

V is a vector perpendicular to the surface, and its length is 
inversely proportional to the normal distance between two con- 
secutive surfaces. In fact (2) shows that v is perpendicular to 
dp, which is any tangent vector, thus proving the first assertion. 
Also, since in passing to a proximate surface we may write 

Svhp = bC, 

we see that F{p + v-'bC) = C+bC. 

This proves the latter assertion. 

It is evident from the above that if (1) be an equipotential, 
or an isothermal, surface, —v represents in direction and magni- 
tude the force at any point or thefltus of heat. And we have seen 

(§ 317) that if 

„ . d , d , d 
dx "^ dy dz 

d^ d^ d^ 

then V = VFp. 

This is due to Hamilton {Lectures on Qimtemions, p. 61 1). 

364. From this it follows that the effect of the vector opera- 
tion V, upon any scalar function of the vector of a point, is to 

M m ^ 


produce the vector which represents in magnitude and direction the 
most rapid change in the value of the function. 

Let us next oonsider the effect of V upon a vector function as 

(r — i^+jri^-K 
We have at once 

and in this semi-Cartesian form it is easy to see that : — 

If o- represent a small vector displacement of a point situated 
at the extremity of the vector p (drawn from the origin) 

SVcr represents the consequent cubical compression of the 
group of points in the vicinity of that considered^ and 

W<r represents twice the vector axis of rotation of the same 
group of points. 


or is equivalent to total differentiation in virtue of our having 
passed from one end to the other of the vector o-. 

365. Suppose we fix our attention upon a group of points 
which originally filled a small sphere about the extremity of p as 
centre, whose equation referred to that point is 

^0) = ^ (1) 

After displacement p becomes p-\-fr, and, by last section, p -f- o) 
becomes p + o + o-— (/ScdV)©-. Hence the vector of the new sur- 
face which encloses the group of points (drawn from the ex- 
tremity of p-f (j) is 

o)i = « — (/Sft)V)(r (2) 

Hence co is a homogeneous linear and vector ftmction of ©i ; or 

and therefore, by (1), 

T<^a>i = e, 

the equation of the new surface, which is evidently a central 
surface of the second order, and therefore, of course, an ellipsoid. 

SECT. 366.] KINEMATICS. 269 

We may solve (2) with great ease by approximation, if we 
remember that T<t is very small, and therefore that in the small 
term we may put Wi for « ; i. e. omit squares of small quan- 
tities; thus, 

0) = «i4-(S'a)iV)o'. 

366. If the small displacement of each point of a medium is in 
the direction of and proportional to, the attraction exerted at that 
point hy any system of material masses, the displacement is effected 
without rotation. 

For if ^ = C be the potential stirface, we have Sadp a com- 
plete differential ; i. e. in Cartesian coordinates 

is a differential of three independent variables. Hence the 
vector axis of rotation 

vanishes by the vanishing of each of its constituents, or 

r.Vcr = 0. 

Conversely, if there be no rotation, the displacements are in the 
direction of and proportional to, the normal vectors to a series of 

For = 7,dpr,V(T =z {8dpV)(T'^VSiTdp, 

where, in the last term, V acts on cr alone. 

Now, of the two terms on the right, the first is a complete 
differential, since it may be written —I)^<t, and therefore the 
remaining term must be so. 

Thus, in a distorted system, there is no compression if 

5V(r = 0, 
and no rotation if 

V,V<T = ; 

and evidently merely transference if <r = a = a constant vector, 

which is one case of 

Vo- = 0. 

270 QdATERNIONa [chap. X. 

In the important case of 

<r = eVFp 
there is evidently no rotation^ since 

V<r = eV'Fp 
is evidently a scalar. In this case, then^ there are only transla- 
tion and compression^ and the latter is at each point proportional 
to the density of a distribution of matter^ which would give 
the potential Fp, For if r be such density^ we have at once 

367. The moment of inertia of a body about a unit vector 
a as axis is evidently 

where p is the vector of the portion m of the mass^ and the 

origin of p is in the axis. 

Hence if we take kTa = e*, we have, as locus of the extremity 

of a. 

Me* =^2m{Fapy = MScufta (suppose), 

the momental ellipsoid. 

If 9 be the vector of the centre of inertia, <t the vector of m 
with respect to it, we have 

p = vr-\-(T; 
therefore Jffif = -2»*{(rai!r)»4-(^a<r)»} 

= -if (Faisr)* '^MSa(t>ia. 

Now, for principal axes, i is max., min., or max.-min, with the 


o« = — 1. 

Thus we have Sa^mFa^—fpio} = 0, 

Saa = ; 

therefore — i^ia+wFaw = joa = i^a (by operating by So), 

Hence (<^i4-** + 'BT»)a = +'BT&it!r, (1) 

• Proc. R,S.E, 1862-3. 


determines the values of a, k^ being found from the equation 

/St!r(<^-fii^ + tsr«)-*i!r = 1 (2) 

Now the normal to 

&(<(>4->t»-fi!r')-><r = 1, (3) 

at the point o- is (<^ + ^* 4- 'fs^Y * ^• 

But (3) passes through — tsr, by (2), and there the normal is 
(<^ + A« + t!r«)-*tsr, 
whichi by (1), is parallel to one of the required values of a. 
Thus we prove Thomson's theorem that the principal axes at 
any point are normals to the three surfaces, confocal with the 
momental ellipsoid , which pass through that point. 


1. Form, from kinematical principles, the equation of the 
cycloid j and employ it to prove the well-known elementary 
properties of the arc, tangent, radius of curvature, and evolute, 
of the curve. 

2. Interpret, kinematically, the equation 

where j3 is a given vector, and a a given scalar. 

Show that it represents a plane curve -, and give it in an 
integrated form independent of^. 

3. If we write 

-car = fit — p, 

the equation in (2) becomes 

P — tsr = aUtff, 
Interpret this kinematically j and find an integral. 

What is the nature of the step we have taken in transform- 
ing from the equation of (2) to that of the present question ? 

272 QUATERNIONS. [chap. X. 

4. The motion of a point in a plane being given, refer it to 

(a.) Fixed rectangular vectors in the plane. 

(d.) Rectangular vectors in the plane^ revolving uniformly 
about a fixed point. 

(c.) Vectors, in the plane, revolving with different, but 
uniform, angular velocities. 

(d,) The vector radius of a fixed circle, drawn to the point 
of contact of a tangent from the moving point. 

In each case translate the result into Cartesian coordinates. 

6. Any point of a line of given length, whose extremities 
move in fixed lines in a given plane, describes an ellipse. 

Show how to find the centre, and axes, of this ellipse ; and 
the angular velocity about the centre of the ellipse of the tracing 
point when the describing line rotates uniformly. 

Transform this construction so as to show that the ellipse 
is a hypotrochoid. 

6. A point, Af moves uniformly round one circular section 
of a cone ; find the angular velocity of the point, a, in which 
the generating line passing through A meets a subcontrary 
section, about the centre of that section. 

7. Solve, generally, the problem of finding the path by which 
a point will pass in the least time from one given point to 
another, the velocity at the point of space whose vector is p 
being expressed by the given scalar function 

Take also the following particular cases : — 
(a.) /p z= a while Sap> 1, 
fp = b while Sap< 1. 
{6.) fp = Sap. 
(c.) fp^-pK 


8. If, in the preceding question, fy be such a ftmction of Tp 
that any one swiftest path is a circle, every other such path 
is a circle, and all paths diverging from one point converge 
accurately in another. 

(Maxwell, Cam. and Bub, Math. Jov/rnaly IX. p. 9.) 

9. Interpret, as results of the composition of successive conical 
rotations, the apparent truisms 

and '^ ^ ^ -^^=1- 

(Hamilton, Lectures^ p. 334.) 

10. Interpret, in the same way, the quaternion operators 

? = (80*(«f-')*(f«-)*> 

-d ?=o*a)*(j)*(i)*(^)*- ii^.) 

11. Knd the axis and angle of rotation by which one given 
rectangular set of unit vectors o, ^3, y is changed into another 
given set a^, ^i, y,. 

12. Show that, if 

the linear and vector operation <f> denotes rotation about the 
vector f, together with imiform expansion in all directions per- 
pendicular to it. 

Prove this also by forming the operator which produces the 
expansion without the rotation, and that producing the rotation 
without the expansion ; and jGinding their joint effect. 

13. Express by quaternions the motion of a side of one right 
cone rolling uniformly upon another which is fixed, the vertices 
of the two being coincident. 

14. Given the simultaneous angular velocities of a body about 

N n 


the principal axes throngli its centre of inertia^ jGind the position 
of these axes in space at any assigned instant. 

15. Find the linear and vector function^ and also the quater- 
nion operator^ by which we may pass, in any simple crystal 
of the cubical system^ from the normal to one g^ven face to that 
to another. How can we use them to distinguish a series of 
fiices belonging to the same zone ? 

16. Classify the simple forms of the cubical system by the 
properties of the linear and vector fiinction, or of the quaternion 

17. Find the vector normal of a face which truncates sym- 
metrically the edge formed by the intersection of two given 

18. Find the normals of a pair of faces symmetrically trun- 
cating the given edge. 

19. Find the normal of a face which is equally inclined to 
three given faces. 

20. Show that the rhombic dodecahedron may be derived 
from the cube, or from the octahedron, by truncation of the 

21. Find the form whose faces replace, symmetrically, the 
edges of the rhombic dodecahedron. 

22. Show how the two kinds of hemihedral forms are indi- 
cated by the quaternion expressions. 

23. Show that the cube may be produced by truncating the 
edges of the regular tetrahedron. 

24. Point out the modifications in the auxiliary vector function 
required in passing to the pyramidal and prismatic systems 

25. In the rhombohedral system the auxiliary quaternion 


operator assumes a singularly simple form. Give this form, and 
point out the results indicated by it. 

26. Show that if the hodograph be a circle, and the accelera- 
tion be directed to a fixed point; the orbit must be a conic 
section, which is limited to being a circle if the acceleration 
follow any other law than that of gravity. 

27. In the hodograph corresponding to acceleration f{I)) di- 
rected towards a fixed centre, the curvature is inversely as 

28. If two circular hodographs, having a common chord, 
which passes through, or tends towards, a common centre of 
force, be cut by any two common orthogonals, the sum of the 
two times of hodographically describing the two intercepted 
arcs (small or large) will be the same for the two hodographs. 
(Hamilton, EleTnents, p. 725.) 

29. Employ the last theorem to prove, after Lambert, that 
the time of describing any arc of an elliptic orbit may be ex- 
pressed in terms of the chord of the arc and the extreme radii 

N n a 



' 868. %j^ Jfi propose to conclade the work by giving a few 
▼ ▼ instances of the ready applicability of quater- 
nions to questions of mathematical physics^ upon which^ even 
more than on the Geometrical or Kinematical applications^ the 
real usefulness of the Calculus must mainly depend— except^ of 
course^ in the eyes of that. section of mathematicians for whom 
Transversals and Anharmonic Pencils^ &c. have a to us incompre- 
hensible charm. Of course we cannot attempt to give examples 
in all branches of physics, nor even to carry very far our in- 
vestigations in any one branch : this Chapter is not intended 
to teach Physics, but merely to show by a few examples how 
expressly and naturally quaternions seem to be fitted for attack- 
ing the problems it presents. 

We commence with a few general theorems in Dynamics — 
the formation of the equations of equilibrium and motion of a 
rigid system, some properties of the central axis, and the motion 
of a solid about its centre of inertia. 

369« When any forces act on a rigid body, the force ft at 
the point whose vector is a, &c., then, if the body be slightly 
displaced, so that a becomes a -h ia, the whole work done is 

This must vanish if the forces are such as to maintain equili- 
brium. Hence tie condition ofequiliirium of a rigid body is 

iSfiba = 0. 


For a displacement of translation ba is any constant vector^ hence 

2)3 = (1) 

For a rotation-displacement, we have by § 349, € being the 
axis, and 2V being indefinitely small, 

ba = Fea, 

and 28.fir€a = I8.€rafi = 8.€2{rafi) = 0, 

whatever be 6, hence 

S.raj3 = (2) 

These equations, (1) and (2), are equivalent to the ordinary six 
equations of equilibrium. 

370. In general, for any set of forces, let„ 
it is required to find the points for which the couple o^ has its 
axis coincident with the resultant /orce fi^. Let y be the vector 
of such a point. 

Then for it the axis of the couple is 

S.r(a-y)j3 = a,-ry^„ 
and by condition 

Operate by Sfi^ ; therefore 

and TyiSi = a,-Pr'Sa,p^ = -fi^Va^fir', 

or y = f^a,pr'+y?u 

a straight line (the Central Axis) pstrallel to the resultant force. 

STL To find the points about which the couple is least. 
Here T{ai — Vyfii) = minimum. 

Therefore S.{a,''Fyfi,)Vp,y = 0, 

278 QUATERNIONS. [chap. XL 

where y' is any vector whatever. It is useless to try /= /3i^ 
but we may put it in succession equal to a^ and Faifii. Thus 

and {Va,p,y-fi]S.yra,p^ = 0. 

Hence y = xFa^p^ +yp^, 

and by operating with S.Faifii, we get 

or y = Fa,pr'+yfii, 

the same locus as in last section. 

372. The couple vanishes if 

This necessitates 8a^ fii = 0^ 

or the force must be in the plane of the couple. If this be the 


still the central asds. 

373. To assign the values of forces ^, ^i, to act at e^ €i, and 
be equivalent to the given system. 

Hence r€$+ Fe, QSi -0 = ^u 

and f = (€-€0-^(a,-r€ii30-»-^(^~O- 
Similarly for ^i. The indefinite terms may be omitted^ as they 
must evidently be equal and opposite. In fact they are any 
equal and opposite forces whatever acting in the line joining 
the given points. 

374. For the motion of a rigid system^ we have of course 
by the general equation of Lagrange. 


Suppose the displacements da to correspond to a mere i^ansla- 
tiony then ha is any constant vector^ hence 

2(m— i3) = 0, 
or, if Oi be the vector of the centre of inertia, and therefore 

we have at once 

and the centre of inertia moves as if the whole mass were con- 
centrated in it^ and acted upon hj all the applied forces. 

375. Again, let the displacements ha correspond to a rotation 
about an axis €, passing through the origin, then 

ha = Fea, 
it being assumed that Te is indefinitely small. 

Hence 'S,8.€Va{ma—^) = 0, 

for all values of 6, and therefore 

2.ra(^a-i3) = 0, 

which contains the three remaining ordinary equations of 

Transfer the origin to the centre of inertia, i. e. put o=ai +«r, 
then our equation becomes 

2r(oi-|-«r)(«wii+«2iT— jS) = 0. 
Or, since "S,in^ = 0, 

2r«r(;»tBr— )8)+rai(ai2«»— 2)3) = 0. 

But ai2«»— 2)3 = 0, hence our equation is simply 

2risr(»^--)3) = 0. 

Now 2FW)3 is the couple, about the centre of inertia, pro- 
duced by the applied forces ; call it 0, then 

^mFfff^ = (f) (1) 

376. Integrating once, 

ImFtff^ = y-\/<l)dt (2) 

280 QUATERNIONS. [chap. XL 

Agaiiij as the motion considered is relative to the centre of 
inertia^ it most be of the nature of rotation about some axis^ 
in general variable. Let c denote at once the direction of^ and 
the angular velocity about^ this axis. Then^ evidently^ 

%r = Few. 
HencCj the last equation may be written 

Operating by S.€, we get 

2«i(rc«r)« = ^cy-hSc/^ (3) 

But^ by operating directly by 2/8€dt upon the equation (1)^ 

we get 

2«i(rw)«=-A« + 2/S€<^^. (4) 

(2) and (4) contain the usual four integrals of the first order. 

877. When no forces act on the body^ we have ^ = 0^ and 


2l»«rFc«r = y, (5) 

2»cr« = 2«l(r€«r)« = -**, (6) 

and^ from (5) and (6)^ 

% = -*« (7) 

One interpretation of (6) is^ that the kinetic energy of rota- 
tion remains unchanged: another is^ that the vector e ter- 
minates in an ellipsoid whose centre is the origin^ and which 
therefore assigns the angular velocity when the direction of the 
axis is given ; (7) shows that the extremity of the instantaneous 
axis is always in a plane fixed in space. 

Also, by (5)^ (7) is the equation of the tangent plane to (6) at 
the extremity of the vector €. Hence the ellipsoid (6) rolh on 
the plane (7). 

From (5) and (6)^ we have at once^ as an equation which e 
must satisfy^ 


This belongs to a cone of the second degree fixed in the body. 
Thus all the ordinary results regarding the motion of a rigid 
body under the action of no forces, the centre of inertia being 
fixed, are deduced almost intuitively : and the only difficulties 
to be met with in more complex cases of such motion are those 
of integration, which are inherent to the subject, and appear 
whatever analytical method is employed. (Hamilton, Proc. 
KLA. 1848.) 

378. We next take Presnel's Theory of Double Refraction, 
bnt merely for the purpose of showing how quaternions simplify 
the processes required, and in no way to discuss the plausibility 
of the physical assumptions. 

Let i/GT be the vector displacement of a portion of the ether, 
with the condition 

^' =-1, (1) 

the force of restitution, on PresnePs assumption, is 

t{aH8iwr-\-6^j8jm-\-c^iSkxsr) = ^^tsr, 
using the notation of Chapter V. Here the function is 
obviously self-conjugate. «% i% c^ are optical constants de- 
pending on the crystalline medium, and on the colour of the 
light, and may be considered as given. 

Fresnel's second assumption is that the ether is incompres- 
sible, or that vibrations normal to a wave front are inadmissible. 
If, then, a be the unit normal to a plane wave in the crystal, 
we have of course 

a'=-l, (2) 

and Sanr = ; (3) 

but, and in addition, we have 

'BT~^V^(f)'Gr II a, 

or Aa-BT*^ = (4) 

This equation (4) is the embodiment of Fresnel's second assump- 
tion, but it may evidently be read as meaning, tke normal to the 
front y the direction of vibration, and that of the force of restitution 
are in one plane ^ 

o o 

282 QUATERNIONS. [chap. XI. 

379. Eqaations (3) and (4)^ if satisfied by m, are also satisfied 
by cra^ so tbat tbe plane (3) intersects the cone (4) in two lines 
at rigbt angles to each other. That is, /or any given wave front 
there are two directions ofvidration^ and thetf are perpendicular to 
each other, 

380. The square of the normal velocity of propagation of a 
plane wave is proportional to the ratio of the resolved part 
of the force of restitution in the direction of vibration, to the 
amount of displacement, hence 

Hence PresnePs Wave-mrface is the envelop of the plane 

Sap = wSm^, (5) 

with the conditions ot«= — 1, (1) 

a»=-l, (2) 

Sam^O, (3) 

S,at!T(t>w = (4) 

Formidable as this problem appears, it is easy enough. From 
(3) and (4) we get at once, 

xxff = F. a VaxftnT, 
Hence, operating by S.rff, 

— ar = — iScr^flj = — y*. 
Therefore (<^ -|- t?*)«r = — a8a(f>'Gr, 

and i8'.a(0+t7«)-»o= (6) 

In passing, we may remark that this equation gives the normal 

velocities of the two rays whose fronts are perpendicular to a. In 

Cartesian coordinates it is the well-known equation 

l^ m} n^ 
1- H = 0. 

By this elimination of «r, our equations are reduced to 

AS.a(0 + i?^)-»a = 0, (6) 

v=:^Sap, (5) 

a^=-l (2) 


They give at once, by § 309, 

(<^ + v2)-ia+«;p/Sa(0 + t;»)-»a = ka. 
Operating by S.a we have 

Substituting for k, and remarking that 

because <^ is self-conjugate, we have 

This gives at once, by rearrangement. 

Hence (0— p2)-ip = """^ . 

Operating by 5.p on this equation we have 

«p(0-p^)-p=-l, (7) 

which is the require^ equation. 

[It will be a good exercise for the student to translate the 
last ten formulae into Cartesian coordinates. He will thus 
reproduce almost exactly the steps by which Archibald Smith ^ 
first arrived at a simple and symmetrical mode of efiecting the 
elimination. Yet, as we shall presently see, the above process 
is far from being the shortest and easiest to which quaternions 
conduct us.] 

SSL The Cartesian form of the equation (7) is not the usual 
one. It is, of course, 

x'^ y^ z"^ 

But write (7) in the form 

or S'P-T^P = 0> 

* Cambridge Phil. Trans. 1843. 
O o il 

284 QUATERNIONS. [chap. XI. 

and we have the usual expression 

a*ap* J«y« c*z^ 

Tliis last quaternion equation can also be put into either of the 
new forms ^ ± 

or y(p-«-<(>-i)-ip = 0. 

382. By applying the results of §§ 171, 172 we may in- 
troduce a multitude of new forms. We must confine ourselves 
to the most simple; but the student may easily investigate 
others by a process precisely similar to that which follows. 

Writing the equation of the wave as 
Sp{(l>-'-^ff)-'P = 0, 
where we have ff = — p-% 

we see that it may be changed to 

if mSp(l>p = ffAp* =— A. 

Thus the new form is 

Sp{<l>-^-mSp(t>p)-'p = 0* (1) 

Here m = ^ ^ , 5/w^p = a'^ar' + iy + c«^% 

and the equation of the wave in Cartesian coordinates is, putting 

383. By means of equation (1) of last section we may easily 
prove Pliicker^s Theorem. Tke Wave-Surface is its own reciprocal 
with respect to the ellipsoid whose equation is 


The equation of the plane of contact of tangents to this surface 
from the point whose vector is p is 

8xn<tt*p = — ^ — • 

The reciprocal of this plane^ with respect to the unit-sphere 
about the origin, has therefore a vector o- where 

Hence p = —. — ^""^or, 

and when this is substituted in the equation of the wave we 
have for the reciprocal (with respect to the unit-sphere) of the 
reciprocal of the wave with respect to the above ellipsoid. 

8.<r{^— J- 5<r^-i<r) or = 0, 

This differs from the equation (1) of last section solely in having 
<^-* instead of ^, and (consistently with this) — instead of m. 

Hence it represents the index-surface. The required reciprocal 
of the wave with reference to the ellipsoid is therefore the wave 

384. Hamilton has given a remarkably simple investigation 
of the form of the equation of the wave-surface, in his Elements 
p. 736, which the reader may consult with advantage. The 
following is essentially the same, but several steps of the process, 
which a skilled analyst would not require to write down, are 
retained for the benefit of the learner. 

Let Slip =-1 (1) 

be the equation of any tangent plane to the wave, i. e. of any 
wave-front. Then p. is the vector of wave-slowness, and the 

normal velocity of propagation is therefore -jjr- . Hence, if -bj be 

the vector direction of displacement, pr^xa \s the effective com- 


ponent of the force of Testitution. Hence, ^o- denoting the 
whole force of restitution, we have 

<^ — fA-»tJ II IJL, 

or V II {<(>-M"')"'M. 

and, as «r is in the plane of the wave-front, 

Sfl'ST = 0, 

or 5/A(<^«)-> = (2) 

This is, in reality, equation (6) of § 380. It appears here, 
however, as the equation of the Index-Surface, the polar reci- 
procal of the wave with respect to a unit-sphere about the 
origin. Of course the optical part of the problem is now solved, 
all that remains being the geometrical process of § 311. 

385. Equation (2) of last section may be at once transformed, 
by the process of § 381, into 

Let us employ an auxiliary vector 

whence ft = 0*'— 0"*)^ (1) 

The equation now becomes 

*tr = 1, (2) 

or, by(l), fi«r«~5T0-*T = 1 (3) 

Differentiating (3), subtract its half from the result obtained by 
operating with S,t on the differential of (1). The remainder is 

T^S^fl — STdfJ, =r 0. 

But we have also (§ 311) 

SpdfA = 0, 

and therefore xp = [it^^t, 

where a? is a scalar. 

This equation, with (2), shows that 

Srp = (4) 


Hence, operating on it by S.p, we have by (1) of last section 

and therefore p"" * = — ^i + t" * . 

This gives p-* = ft^— t"*. 

Substituting from these equations in (1) above, it becomes 
T-'^p-' = (p-« + T-«-(/>-Or, 
or T = (<^"''— P"*)~V~*- 
Finally, we have for the required equation, by (4), 
8p-'{(l>-^—p-^)-^p-' = 0, 
or, by a transformation already employed, 

Sp{4^^p^r'p=:- 1. 

386. It may assist the student in the practice of quaternion 
analysis, which is our main object, if we give a few of these 
investigations by a somewhat varied process. 

Thus, in § 378, let us write as in § 168, 

a^iSivf-Ji-b^jSjfff + c^kSkm = \'8f/vr+f/8\f'ST''jf/tir. 

We have, by the same processes as in § 378, 

8.vray8i/isT-{-8.'UTapf8K'isr = 0. 

This may be written, so Jhr as the generating lines we require are 

8.maVX'mp! = = 8.'usa}^'STik, 
since 'ora is a vector. 
Or we may write 

8.f/F.isT\'^a = = S.f/tffy'BTa. 

Equations (1) denote two cones of the second order which pass 
through the intersections of (3) and (4) of § 378. Hence their 
intersections are the directions of vibration. 

387. By (1) we have 

8,vrK^iffafx = 0. 




[chap. XI. 

Hence vX V, a, y! are coplanar ; and, as «r is perpendicnlar to 
a, it is equally inclined to V\'a and VyLO, 

For, a L, M, A be the 
A projections of X', fi, a on 

the unit sphere, £C the 
great circle whose pole is 
A, we are to find for the 
projections of the values of 
«r on the sphere points F 
and P', such that if LP be 
produced till PQ = Zp, Q 
may lie on the great circle 
AM. Hence, evidently, 
^ CP ^ PB, 

and UP^ = PB', which proves the proposition, since the pro- 
jections of FX'a and Viia on the sphere are points b and c in EC, 
distant by quadrants from C and B respectively. 

388. Or thus, 8ma = 0, 

S.^F.a\'vrfi = 0, 
therefore xw = F.aF.aX'«r/ui', 

= - VX^i/'-aSaFXr^Tf/. 
Hence {SK'i/'-x)^ = (V + o5aA')^M«^ + (M +a/Safi')S'AV. 
Operate by SX, and we have 

= Sf/rsrT^FK'a. 
Hence by symmetry. 




= 0, 

and as 

TFxfa - TFf/a 

^wa =0 , 
iir= U{Urk'a±Uri/a). 


389. The optical interpretation of the common result of the 
last two sections is that tie planes of polarization of the two rays 
whose wave^fronts wre parallel, bisect the angles contained by planes 
passing through the normal to the wave-front and the vectors (optic 
axes) A', ii\ 

390. As in § 380, the normal velocity is given by 

V* = 8m(^ = 2 Sk'taSiirff—p'^* 

[This transformation, effected by means of the value of «r in 

§ 388, is left to the reader.] 

Hence, if t?i, t?,, be the velocities of the two waves whose 

normal is a 

t?;-r; = 2T.rK'aVfxa 

oc sinX'asin/ji'a. 
That is, the difference of the squa/res of the velocities of the two 
waves varies as the product of the sines of the angles between the 
normal to the wave-front and the optic axes (X', ft'). 

39L We have, obviously. 

Hence t;« = / q: {T± S). FxTa Ff/a. 

The equation of the index surface, for which 

l}, = i, Up = a, 

is therefore 

1 = -//)« ip {T± 8). Fk'p Ff/p. 

This will, of course, become the equation of the reciprocal of the 
index-surface, i. e. the wave-surface, if we put for the fimction 
^ its reciprocal : i. e. if in the values of X', /, p^ we put 

- , -7 , - {or a,b,c respectively. We have then, and indeed 


290 QUATERNIONS. [chap. XI 

it might have been deduced even more simply as a transfonna- 
tion of § 380 (7), 

1 =^j>p'^:{T±S).r\priip, 

as another form of the equation of FresneFs wave. 

If we employ the i, k transformation of § 121, this may be 
written^ as the student may easily prove^ in the form 

(K«-.i«)« = 5>(i-ic)p+(rrip+rricp)». 

392. We may now^ in furtherance of our object^ which is to 
give varied examples of quaternions^ not complete treatment of 
any one subject^ proceed to deduce some of the properties of the 
wave-surface from the different forms of its equation which we 
have given. 

393. FremeVa comtruction of tie wave by points. 

From § 273 (4) we see at once that the lengths of the 
principal semidiameters of the central section of the ellipsoid 

SfX^'^p = 1, 

by the plane Sap =s 0^ 

are determined by the equation 

Aa(<^-*-p-»)-'a = 0. 
If these lengths be laid off along a, the central perpendicular to 
the cutting plane^ their extremities lie on a surface for which 
a = Up, and Tp has values determined by the equation. 
Hence the equation of the locus is 

as in §§ 380^ 385. 

Of course the index-surface is derived firom the reciprocal 
eUipsoid Sp<l>p = 1 

by the same construction. 

394. Again^ in the equation 

1 =-pp'+{T±s).r\prpLp, 


suppose V\p = 0, or Viip = 0, 

we obviously have 

p= ±-— or p=: ±--p> 
Vp vp 

and there are therefore four singular points. 

To find the nature of the sur£EU3e near these points put 


where T'gt is very small^ and reject terms above the first order in 
T'ur, The equation of the wave becomes, in the neighbourhood 
of the singular point, 

which belongs to a cone of the second order. 

395. From the similarity of its equation to that of the wave, 
it is obvious that the index-surface also has four conical cusps. 
As an infinite number of tangent planes can be drawn at such 
a point, the reciprocal surface must be capable of being touched 
by a plane at an infinite number of points ; so that the wave- 
surface has four tangent planes which touch it along ridges. 

To find their form, let us employ the last form of equation of 
the wave in § 39 1 . If we put 

TTip = TFkp, (1) 

we have the equation of a cone of the second degree. It meets 
the wave at its intersections with the planes 

5(i-K)p=± (««-.>) (2) 

Now the wave-surface is Umched by these planes, because we 
cannot have the quantity on the first side of this equation 
greater in absolute magnitude than that on the second, so long 
as p satisfies the equation of the wave. 

That the curves of contact are circles appears at once from 
(1) and (2), for they give in combination 

p» = +5(t+K)p, (3) 

Pp 5& 

292 Q0ATRRNIONS. [chap. XI. 

the eqofttioiiB of two spheres on which the carves in question are 

The diameter of this drcoLir ridge is 

TF.(i+^)U{i-K) = ^^j = l^/(a.-4.)(«•-c«). 

[Simple as these processes are^ the student will find on trial 
that the equation 

gives the results quite as simply. For we have only to examine 
the cases in which — p'* has the value of one of the roots of the 
symbolical cubic in ^"^^ In the present case Tp^6 is the only 
one which requires to be studied.] 

896. By § 384^ we see that the auxiliary vector of the suc- 
ceeding section^ viz. 

is parallel to the direction of the force of restitution^ (fyar. Henoe^ 
as Hamilton has shown^ the equation of the wave^ in the form 

(4) of § 385^ indicates that tAe direction of tie force of restitu^ 
tion is perpendicular to tie ray. 

Again^ as for any one versor of a vector of the wave there are 
two values of the tensor^ which are found from the equation 

S.Up{il>-''-'P'^)-^Up = 0, 
we see by § 393 that tAe lines of vibration for a given plane 
front are parallel to the axes of any section of the ellipsoid 

S.p<l>-^p = 1 
made by a plane parallel to the front ; or to the tangents to the 
lines of curvature at a point where the tangent plane is parallel 
to the wave-front. 

397. Again, a curve which is drawn on the wave-surface so as 
to touch at each point the corresponding line of vibration has 


Hence iS^pdp = 0, or 8p<l>p = C, 

so that such curves are the intersections of the wave with a 
series of ellipsoids concentric with it. 

398. For curves cutting at right angles the lines of vibration 
we have ^^ y Fp<f>-' {<!>-' --p-^y^p 

II rp(4^-pT'p' 

Hence 8pdp = 0, or Tp = C, 

so that the curves in question lie on concentric spheres. 
They are also spherical conies, because where 
Tp = C 
the equation of the wave becomes 

5.p(^-* + C-«)-v = o, 
the equation of a cyclic cone^ whose vertex is at the common 
centre of the sphere and the wave-surface^ and which cuts them 
in their curve of intersection. 

399* As a final example we take the case of the action of 
electric currents on one another or on magnets; and the mutual 
action of permanent magnets. 

A comparison between the processes we employ and those of 
Ampere {Theorie des Phenomenes Electrodynamiques, 8fc,, many 
of which are well given by Murphy in his Electricity) will at 
once show how much is gained in simplicity and directness by 
the use of quaternions. 

The same gain in simplicity will be noticed in the investiga- 
tions of the mutual effects of permanent magnets^ where the 
resultant forces and couples are at once introduced in their most 
natural and direct forms. 

400. Ampere's experimental laws may be stated as follows : 

I. Equal and opposite currents in the same conductor produce 
equal and opposite effects on other conductors : whence it follows 


that an dement of one corrent lias no effect on an element of 
another which lies in the plane bisecting the former at right 

II. The effect of a conductor bent or twisted in any manner 
is equivalent to that of a straight one, provided that the two 
are traversed by equal currents, and the former nearly coincides 
with the latter. 

III. No closed circuit can set in motion an element of a cir- 
cular conductor about an axis through the centre of the circle 
and perpendicular to its plane. 

rV. In similar systems traversed by equal currents the forces 
are equal. 

To these we add the assumption that the action between two 
elements of currents is in the straight line joining them : and 
two others, viz. that the effect of any element of a current on 
another is directly as the product of the strengths of the cur- 
rents, and of the lengths of the elements. 

40L Let there be two closed currents whose strengths are 
a and a^ ; let a, a^ be elements of these, a being the vector 
joining their middle points. Then the effect of a on a^ must, 
when resolved along a^, be a complete differential with respect 
to a (i. e. with respect to the three independent variables in- 
volved in o), since the total resolved effect of the closed circuit 
of which a' is an element is zero by III. 

Also by I, II, the effect is a function of Ta, 8aa% Saui, and 
Sa^ai, since these are sufficient to resolve a and a^ into elements 
parallel and perpendicular to each other and to a. Hence the 
mutual effect is 

aa,Ua/{Ta, Sua, Saa,, Saa,), 

and the resolved effect parallel to a^ is 
aaiSUai Uaf. 


AIso^ that action and reaction may be equal in absolute mag- 
nitude^ y must be symmetrical in Sojol and Saa^, Again^ o( (as 
differential of a) can enter onh/ to the f/rst power, and mmt appear 
in each term of/*. 

Hence / = ASaa^^ + BSaaSaa^. 

But, by rV, this must be independent of the dimensions of the 
system. Hence -4 is of —2 and 5 of —4 dimensions in Ta. 

-^ {ASaa^Sa'a^ + BSaaS^aa^} 
is a complete differential, with respect to a, if £?a = a\ Let 

A- ^ 
where C is a constant depending on the units employed, there- 
fore d^ = ^Saa', 

and the resolved effect 

The factor in brackets is evidently proportional in the ordinary 

notation to 

sin ^ sin ^ cos o)— ^ cos ^ cos ^ 

402. Thus the whole force is 

Caa^a ^S^aa^ ^ Caa^a ^ S^aa' 

as we should expect, d^a being = a^. [This may easily be 
transformed into 

or 5 = f , 

2 Coat Ua „ ,„ , . 
-. — ddi (To)*, 


which is the quaternion expression for Ampere's well-known 

408. The whole effect on Oi of the closed circuity of which tk 
is an element^ is therefore 

Guit f a ASaaiY 
2 J 8aa^ To» ' 

^ CcMi e aSaai ^ f VnxC ^ 

between proper limits. As the integrated part is the same at 
both limits^ the effect is 

and depends on the form of the closed circuit. 

404. This vector p, which is of great importance in the 
whole theory of the effects of closed or indefinitely extended 
circuits^ corresponds to the line which is called by Ampere 
*'directrice de Inaction Slectrodynamique/* It has a definite value 
at each point of space^ independent of the existence of any other 

Consider the circuit a polygon whose sides are indefinitely 
small; join its ang^ar points with any assumed pointy erect 
at the latter^ perpendicular to the plane of each elementary 

triangle so formed^ a vector whose length is -^ where » is the 

vertical angle of the triangle and r the length of one of the 
containing sides ; the sum of such vectors is the '^ directrice'^ at 
the assumed point. 

405. The mere form of the result of § 403 shows at once 
that if the element ax be turned about its middle point, the direction 
of the resultant action is confmed to the plane whose normal is fi. 

Suppose that the element a^ is forced to remain perpendicular 
to some given vector b, we have 


and the whole action in its plane of motion is proportional to 

But r.hVa^p -^a^S^b. 

Hence the action is evidentlj constant for all possible positions 
of ai; or 

The effect of any system of closed currents on an element of a 
conductor which is restricted to a given jplane is {in that plane) 
independent of the direction of the element, 

406. Let the closed current heptane and very small. Let € 
(where IV = I) be its normal^ and let y be the vector of any 
point within it (as the centre of inertia of its area); the middle 
point of ai being the origin of vectors. 

Let o = y+p; therefore d = p', 

and ^ = fIS± = fl(y±M 

to a sufficient approximation. 
Now (between limits) 

fFpp' = 2^6, 
where A is the area of the closed circuit. 
Also generally 

/Fyp'Syp = HSypryp+yF.y/Fpp') 
= (between limits) Ay Vy€, 
Hence for this case 

407. If> instead of one small plane closed current, there be 
a series of such, of equal area, disposed regularly in a tubular 


form^ let x be the distance between two consecutive currents 
measured along the axis of the tube; then^ putting y = x€, 
we have for the whole effect of such a set of currents on at 


Va f(^ . 3y£yy\ 

= — — - -=^ (between proper hmits). 

If the axis of the tubular arrangement be a closed curve this 
will evidently vanish. Hence a closed solenoid exerts no influence 
on an element of a conductor. The same is evidently true if tie 
solenoid be indefinite in both directions. 

If the axis extend to infinity in one direction^ and yo be the 
vector of the other extremity^ the effect is 

CAaa^ Fuiyo 

'~2x Tyl ' 
and is theretore perpendicular to the element and to the line joining 
it with the extremity of the solenoid. It is evidently inversely 
as Tyl and directly as the sine of the angle contained between the ^ 
direction of the element and that of the line joining the latter with 
the extremity of the solenoid. It is also inversely as Xy and there- 
fore directly as the number of currents in a unit of the axis of the 

408. To find the effect of the whole circuit whose element is 
ai on the extremity of the solenoid^ we must change the sign of 
the above and put a^ = yo'; therefore the effect is 

__ CAaa, C Vy^y, 
2x J 2Vo' ' 

an integral of the species considered in § 403 whose value is 
easily assigned in particular cases. 

409* Suppose the conductor to be straight, and indefinitely 
extended in both directions. 

Let M be the vector perpendicular to it from the extremity of 
the canal^ and let the conductor be || t], where ^ = 21; = I. . . 

BHCT. 410.] 



Therefore y« = Ad+yri (where y is a scalar), 

and the integral in § 407 is 

hVnO / y ^ z. Yy%Q 

The whole effect is therefore 




and is thus perpendicular to the plane passing tirouffk tie con- 
ductor and the extremity of the canal, and varies inversely as the 
distance of the latter from the conductor. 

This is exactly the observed effect of an indefinite straight 
current on a magnetic pole^ or particle of free magnetism. 

410. Suppose the con- 
ductor to be circular, and 
the pole nearly in its axis. 

Let EPD be the con- 
ductor^ AB its axis^ and 
C the pole; BC perpen- 
dicular to AB, and small 
in comparison with AE^h 
the radius of the circle. 

ABhea^i, BC =: bi, 
AP = hijx^-hy) 


CP ^ y ^ a^i-\-bk'~'h{jx-\'ky). 

And the effect on 


2 [ff{{A-by)i-\-a^ixy-\'a^yk} 
a fl I — rs — 1 

where the integral extends to the whole circuit. 


41L Suppose in particular (7 to be one pole of a small 
magnet or solenoid CC whose length is 2 /^ and whose middle 
point is at 6 and distant a from the centre of the conductor. 
Let L CGB = A. Then evidently 

a^ z= a + ^oosA, 
b = ^sinA. 
Also the effect on Cbeoomes^ if a; + d'+^' = ^% 

2;y^{(*-*y)*+«i^+«iy*}{i -^-jT + Y ~^ ■'■•••^ 

since for the whole circuit 

/(Ty- = 2ir 


/(Toy- = 0. 
If we suppose the centre of the magnet fixed, the vector axis 
of the couple produced by the action of the current on C is 

ir.{i cos A + * sin ^)f^ 

TtA^lsinA .r Sb* 15 A*J* Ba^b cos A . 


j8 J\ A^ ^ 2 A' A* sin A 

If ^, &c. be now developed in powers of I, this at once becomes 
ir^'^sinA »fg 6g^cosA 15a'^'cos«A 3^» 

S^^sin'A 15 A*^'sin*A (a + ^co8A)^cosA • 5a?cosAxn 
"" a«+A» "*"2" {a^ + A^y ~ dj«~+>i» ^ tf» + A« )^ ' 

Putting — ^ for ^ and changing the sign of the whole to get 
that for pole (T, we have for the vector axis of the complete 

4TrA»;sinA.. „ ^*(4<^'--A«)(4 — 5 sin«A) „ i 

-(^rT^-^{^+* (^Tk^- ""^-i 


which is almost exactly proportional to sinA if 2ei ^ ^ and I 
be small. 

On this depends a modification of the tangent galvanometer. 
(Bravais — Ann. de Chimiey xxxviii. 309.) 

412. As before^ the effect of an indefinite solenoid on a^ is 

Now suppose Oi to be an element of a small plane circuity d the 
vector of the centre of inertia of its area, the pole of the solenoid 
being origin. 

Let y = 5+/>, then ai = />'. 

The whole effect is therefore 

CAaa^ r 

" 2xlV Vi+~lv">'' 

where A^ and 6i are, for the new circuit, what A and € were for 
the former. 

Let the new circuit also belong to an indefinite solenoid, and 
let do be the vector joining the poles of the two solenoids. 
Then the mutual effect is 

CAA^aa^ f. b' 3bSbb\ 
2xx^ J^Tb^ '^ Tb' ) 

_ CAAiOai bo Ubp 

- 2xx, {TboY '^' i^oY ' 
which is exactly tAe mutual effect of two magnetic poles. Two 
Jmite solenoids J therefore^ act on each other exactly as two mxtgnetSy 
and the pole of an indefinite solenoid acts as a particle of free 

413. The mutual attraction of two indefinitely small plane 
closed circuitsji whose normals are € and €i> may evidently be 



[chap. XL 

dedaced by twice differentiatiiig the expression -^^ for the 

matoal action of the poles of two indefinite solenoids^ making 
d6 in one differentiation || c and in the other || Ct* . 

Bnt it may also be calculated directly by a process which will 
give OS in addition the CQuple impressed on one of the circuits 
by the other, supposing for simplicity the first to be circular. 

Let A and JB be the centres of inertia of the areas of A and B, 
c and Ci vectors normal to their planes^ <r any vector radius of 

Then whole effect on a', by §§ 406, 403, 

A f 3(ff^<r)504-<r)e . 

"" TO^i^^'V^ + tot) -*■ —mi—v + -7^) 


-r Jjg, T^- y^. 

But between proper limits, 

fVi/riSetr = -^, r.jyrSei, 
for generally /Van S»<t = - i ( ri7(r AW<r + T. »y F. d/r<r<TO. 

Hence, after a reduction or two, we find that the whole force 
exerted by A on the centre of inertia of the area of B 

This, as already observed, may be at once found by twice 


differentiating -^=^ . In the same way the vector moment^ due 
to Aj about the centre of inertia of B, 

These expressions for the whole force of one small magnet 
on the centre of inertia of another^ and the couple about the 
latter^ seem to be the simplest that can be given. It is easy to 
deduce from them the ordinary forms. For instance, the whole 
resultant couple on the second magnet 

OC ^ '■> 

may easily be shown to coincide with that given by Ellis 
(Camb. Math, Journaly iv. 95), though it seems to lose in sim- 
plicity and capability of interpretation by such modifications. 

414. The above formulae show that the whole force exerted 
by one small magnet M, on the centre of inertia of another m^ 
consists of four terms which are, in order, 

1st. In the line joining the magnets, and proportional to the 
cosine of tieir muttud inclination. 

2nd. In the same line^ and proportional to five times the product 
of the cosines of their respective inclinations to this Hne. 

3rd. and 4th. Parallel to {-..} anc( proportional to the cosine 


of the inclination of { ^to the joining line. 

All these forces are, in addition, inversely as the fourth power 
of the distance between the magnets. 


For the oonples about the centre of inertia of m we have 

1st. J couple whose axis is perpendicular to each magnet, and 
which is as the sine of their mutual inclination. 

2nd. A couple whose axis is perpendicular to m and to the line 
joining the magnets, and whose moment is as three times the product 
of the sine of the inclination ofxn, and the cosine of the inclination 
ofTAjto the joining line. 

In addition these couples vaiy inversely as the third power of 
the distance between the magnets. 

[These results afford a good example of what has been called 
the internal nature of the methods of quaternions^ reducing^ as 
they do at once^ the forces and couples to others independent 
of any lines of reference^ other than those necessarily belonging 
to the system under consideration. To show their ready appli- 
cability^ let us take a Theorem due to Grauss.] 

415. If two small magnets be at right angles to each other, the 
moment of rotation of the first is approximately twice as great 
when the axis of the second passes through the centre of the fi/rst, as 
when the axis of the first passes through the centre of the second. 

In the first case ^tl/3-^Cx; 

C 2C" 

therefore moment = •mQ;T{€€^'-Z€€^ =s -^^Tcci. 

In the second Cill/9-^c; 

therefore moment = -^j^ Tt€^. Hence the theorem. 


416. Again, we may easily reproduce the results of § 413, if 
for the two small circuits we suppose two small magnets 
perpendicular to their planes to be substituted, fi is then the 
vector joining the middle points of these magnets, and by 
changing the tensors we may take 26 and 2€i as the vector 
lengths of the magnets. 


Hence evidently the mutual effect 

which is easily reducible to 

as before, if smaller terms be omitted. 

If we operate with Kci on the two first terms of the unre- 
duced expression, and take the difference between this result 
and the same with the sign of Ci changed, we have the whole 
vector axis of the. couple on the magnet 2€i, which is therefore, 
as before, seen to be proportional to 

4 , . sn.fiSfie^ 

417. We might apply the foregoing formulae with great ease 
to other cases treated by Ampdre, De Montferrand, &c.— or to 
two finite circular conductors as in Weber's Dynamometer — ^but 
in general the only difficulty is in the integration, which even in 
some of the simplest cases involves elliptic functions, &c., &c. 

418. Let F{y) be the potential of any system upon a unit 
particle at the extremity of y. 

^(y) = C (1) 

is the equation of a level surface. 

Let the differential of (1) be 

Svdy^O, (2) 

then r is a vector normal to (1), and is therefore the direction of 
the force. 

But, passing to a proximate level surface, we have 

Svby = bC. 
Make by = xv^ then 

-xTv' = bC, 
R r 

306 QUATEBNI0K8. [cHAP. XI. 

Henoe v exiwesses the force in magnitude also. (§ 363.) 
Now by § 406 we have for the vector force exerted by a small 
plane closed circuit on a particle of free magnetism the ex- 

omitting the fieustors depending on the strength of the current 
and the strength of magnetism of the particle. 
Hence the potential^ by (2) and (1), 

area of circuit projected perpendicular to y 

a solid angle subtended by circuit. 

The constant is omitted in the integratioD^ as the potential must 
evidently vanish for infinite values of Ty, 

By means of Ampere's idea of breaking up. a finite circuit into 
an indefinite number of indefinitely small ones^ it is evident that 
the above result may be at once extended to the case of such a 
finite closed circuit. 

419. Quaternions give a simple method of deducing the 
well-known property of the Magnetic Curves, 

Let Ay A' be two equal magnetic 
poles^ whose vector distance^ 2a^ is 
bisected in 0, QQ' an indefinitely 
small magnet whose length is 2p, 
where p = OP, Then evidently, 
taking moments, 

F(p + a)p' _ r {p-a)p' 

Tlp^ay - ^ T{p-ay 


Operate by 45. Top, 

or S.aF(-^)U{p + a) = + {same with -a}, 

i. e. SadU{p + a)= ± Sad U{p^a), 

8a{U{p + a)T U{p—a)} = const., 
or cos AOAF ± AOA'P = const., 

the property referred to. 

420. If the vector of any point be denoted by 

p — ix-^jy+kz, (1) 

there are many physically interesting and important transforma- 
tions depending upon the effects of the quaternion operator 

'=4*4^4 <^' 

on various Unctions of p. When the Amotion of p is a scalar, 
the effect of V is to give the vector of most rapid increase. Its 
effect on a vector function is indicated briefly in § 364. 

42L We commence with one or two simple examples, whic£ 
are not only interesting, but very useful in transformations. 

Vp = (i +&c.)(Mr + &C.)=-3 (3) 

V (?»• = n{Tpf-^VTp = n{TpY-'p ; (5) 

and, of course, v^=-^^^; (5)' 

whence, v^ = _^ = -^, (6) 


and, of course, V-i-=-V^ = (6)» 

Also, Vp=z—3 = TpVUp+VTp.Up= TpVUp—l, 

.'. ^Up = -±- (7) 

422. B7 the help of the above results^ of whioh (6) is 
especially useful (though obvious on other grounds)^ and (4) 
and (7) very remarkable^ we may easily find the effect of v upon 
more complex functions. 

Thus, VSap=-v(«P + &c.)=-a, (1) 

Vra/>=-Vrpa=-.V(^-.&ip)= 3a-a = 2a (2) 


--Tap ^ 2a_ ^ SpFap _ 2ap^'\'3pFap ^ ap^ — SpSap . 

„ o^rr'^ap p*SaZp—SSapSphp Sahp SSapSpbp 

Hence ^.«pV — = j,- =^_-___ 

=->w « 

This is a very useAil transformation in various physical appli- 
cations. By (6) it can be put in the sometimes more convenient 

S.ipV^^iS.aV-L. (5) 

And it is worthy of remark that, as inay easily be seen, — iS may 
be put for Fin the left-hand member of the equation. 

423. We have also 
VF^py=V{pSyp^p8Py-{-ySfip} = ^yfi^SS^^Py=Sfiy. (1) 
Hence, if ^ be any linear and vector Amotion of the form 

(l>p=:a + :EFPpy + mp, : (2) 

i. e. a self-conjugate function with a constant vector added, then 
V<^p = 25/3y— 3 1» = scalar (3) 


Hence^ an integral of 

V<r = scalar constant, is <r = <^p (4) 

If the constant value of Vo- contain a vector part, there will 
be terms of the form Fep in the expression for <r, which will then 
express a distortion accompanied by rotation. (§ 366.) 

Also, a solution of Vq = a (where q and a are quaterjiions) is 

q = SCp+F€p + <f>p. 
It may be remarked also, as of considerable importance in 
physical applications, that, by (1) and (2) of § 422, 

but we cannot here enter into details on this point. 

424. It would be easy to give many more of these trans- 
formations, which really present no difficulty ; but it is sufficient 
to show the ready applicability to physical questions of one or 
two of those already obtained ; a property of great importance, 
as extensions of mathematical physics are far more valuable than 
mere analytical or geometrical theorems. 

Thus, if 0- be the vector-displacement of that point of a homo- 
geneous elastic solid whose vector is /», we have, p being the 
consequent pressure produced, 

Vjo-hV»<r = 0, (1) 

whence SbpV^a- =z^8bpVp = bp, a complete diflferential. ... (2) 

Also, generally, p = kSVir, 

and if the solid be incompressible 

/SV<r= (3) 

Thomson has shown {Canii» and Dvh. Math, Journal ^ ii. p. 62), 
that the forces produced by given distributions of matter, elec- 
tricity, magnetism, or galvanic currents, can be represented at 
every point by displacements of such a solid producible by ex- 
ternal forces. It may be usefiil to give his analysis, with some 
additions, in a quaternion form, to show the insight gained by 
the simplicity of the present method. 


425. Thus, if Sahp = h -=- ^^ "^ij write, each equal to 

This gives <r=:-V— , 

the vector-force exerted by one particle of matter or free elec- 
tricity on another. This value of tr evidently satisfies (2) and (3). 
Again, if 

S.hpVa = d ^rr > fitter is equal to 

-S.hpV^^ by (4) of §422. 

Here a particular case is 


which is the vector-force exerted by an element a of a current 
upon a particle of magnetism at p. (§ 407.) 

436. Also, by § 422 (3), 

_ Vap ^ ap* — SpSap 

and we see by §§ 406, 407 that this is the vector-force exerted 
by a small plane current at the origin (its plane being perpen- 
dicular to a) upon a magnetic particle, or pole of a solenoid, at p. 
This expression, being a pure vector, denotes an elementary 
rotation caused by the distortion of the solid, and it is evident 
that the above value of <r satisfies the equations (2), (3), and the 
distortion is therefore producible by external forces. Thus the 
eifect of an element of a current on a magnetic particle is ex- 
pressed directly by the displacement, while that of a small closed 
current or magnet is represented by the vector-axis of the rota- 
tion caused by the displacement. 

427. Again, let 


It is evident that o- satisfies (2)^ and that the right-hand side of 
the above equation may be written 

Hence a particular case is 

and this satisfies (3) also. 

Hence the corresponding displacement is producible by external 
forces, and Vo- is the rotation axis of the element at p, and is 
seen as before to represent the vector-force exerted on a particle 
of magnetism at p by an element a of a current at the origin. 

428. It is interesting to observe that a particular value of a- 
in this case is 

as may easily be proved by substitution. 
Again, if Sbptr =—^-^y 

we have evidently o- = V -^ . 

Now, as ^=-j is the potential of a small magnet a, at the 

origin, on a particle of free magnetism at p, o- is the resultant 
magnetic force, and represents also a possible distortion of the 
elastic solid by external forces, since V<r = V*<r = 0, and thus 
(2) and (3) are both satisfied. 



1. The expression 

denotes a vector. What vector ? 

2. If two Burfftces intersect along a common line of curvature^ 
they meet at a constant angle. 

3. By the help of the quaternion formulae of rotation^ trans- 
late into a new form the solution (given in § 234) of the problem 
of inscribing in a sphere a closed polygon the directions of 
whose sides are given. 

4. Express^ in terms of the masses^ and geocentric vectors 
of the sun and moon^ the sun's disturbing force on the moon^ 
and expand it to terms of the second order ; pointing out the 
magnitudes and directions of the separate components. 

(Hamilton^ Lectures, p. 615.) 

6. If J = r*, show that 

2dq = 2dr^ = i{dr+Kqdrf^)8^^ = ^{dr4-r^drKq)Sf-^ 

= {drq+Kqdr)q-'{q-^Kq)-^ = {drq+Kqdr){r'^li'y' 

_ dr+ Uj-^drUq-'^ _ drUq+ Uq-^dr _ q-^jUgdr-hdrUq-^) 
" Tq{Uq-\-Uq-') ^ q{Uq-\-Uq-') " Uq-\-Uq-^ 

q-^{qdr-\-Trdrq'^) _ drUq+Uq-^dr _ drKf^^+q-^dr 

= {dr + F.Fdr^q}q-' =z {dr ^F.Fdr^q''}q-' 

dr j^^dr F dr j^^^dr F ^ 

= — H-r.r ^q- FF ^q-' 

q q 8^ q q 8^ 

^drq-'-^^FFq-'Fdr^l +-^q-'): 


and give geometrical interpretations of these varied expressions 
for the same quantity. {Ibid. p. 628.) 

6. Derive (4) of § 92 directly from (3) of § 91. 

7. Find the successive values of the continued fraction 


where i and J have their quaternion significations^ and x has the 
values 1, 2, 3, &c. {Lectures, p. 645.) 

8. If we have 

where c is a given quaternion^ find the successive values. 

For what values of c does u become constant? {Ibid. 
p. 652.) 

9. What vector is given, in terms of two known vectors, by 
the relation 

Show that the origin lies on the circle which passes through 
the extremities of these three vectors. 

10. What problem has its conditions stated in the following 
six equations, from which f , 17, C are to be determined as scalar 
ftmctions of x, y, z, or of 

p = ix+jy-^iz? 

V«^ = 0, V^ri = 0, V^C = 0, SV^Vri = 0, 

S^riVC = 0, «Vf V^ = 0, 

1 „ , d , d , d 

where V = ^^ +j^+i-^-. 

Show that (with a change of origin) the general solution of 
these equations may be put in the form 

where <^ is a self-conjugate linear and vector function, and ^, 77, ( 

s s 


are to be found respectively- from the three values oif at any 
point by relations similar to those in Ex. 24 to Chapter IX. 
(See Lam^^ Journal de Mathematiques, 1843.) 

11. Hamilton^ Buhop Laui^s Premium Hxamination, 1862. 

(a.) If OABP be four points of space^ whereof the three 
first are given^ and not collinear ; if also oa = a, 
OB =s p,ov =z p ; and if, in the equation 

a a 

the characteristic of operation F be replaced by S, 
the locus of F is a plane. What plane ? 

(i.) In the same general equation, if jP be replaced by ^ 
the locus is an indefinite right line. What line ? 

(c.) If ^ be changed to K, the locus of p is a point. 
What point? 

(d.) If jPbe made = Uj the locus is an indefinite half-line, 
or ray. What ray ? 

(e,) If ^ be replaced by T^ the locus is a sphere. What 
sphere ? 

{/,) I{ F he changed to TF, the locus is a cylinder of 
revolution. What cylinder ? 

(^.) If JF be made TFU, the locus is a cone of revolution. 
What cone ? 

(A.) If SU be substituted for Fy the locus is one sheet of 
such a cone. Of what cone ? and which sheet ? 

(i.) If F be changed to FU, the locus is a pair of rays. 
Which pair ? 

12. {Ibid. 1863.) 

(a.) The equation Spp' + a^ =z 


expresses that p and p' are the vectors of two points 
p and p', which are conjugate with respect to the 

p» + «» = 0; 

or of which one is on the polar plane of the other. 

( j.) Prove by quaternions that if the right line pp', connect- 
ing two such points^ intersect the sphere, it is cut 
harmonically thereby. 

(<?.) If p' be a given external point, the cone of tangents 
drawn from it is represented by the equation, 

and the orthogonal cone, concentric with the sphere, 
ty (%')'+«V = o- 

(rf.) Prove and interpret the equation, 

T{np-^a) = T(p-na)y if Tp = Ta. 

(e,) Transform and interpret the equation of the ellipsoid, 
r(ip-hpic) = K»-t«. 

(/.) The equation 

expresses that p and p' are values of conjugate 
points, with respect to the same ellipsoid. 

(ff.) The equation of the ellipsoid may also be thus written, 
Svp = 1, if {k^^i^Yv = (i^Kyp + 2iSKp-\-2KSip. 

(A,) The last equation gives also, 

(jc»-4«)»y = (t»-}-JC»)p»-h2FipK. 

(i.) "With the same signification of v, the differential equa- 
tions of the ellipsoid and its reciprocal become 

Svdp = 0, Spdv = 0. 
{J,) Eliminate p between the four scalar equations. 
Sap = a, Spp = 6, 8yp = c, Sep = e, 

SB 7, 


13. {Ibid. 1864.) 

(a.) Let AiBi, A^B^j., .A^B^ ^ "^7 graven system of posited 
right liaes^ the 2n points being all given; and let 
their vector snm^ 

AB =r A,B,-\-A^B^-\-,.. + AJB^ 

be a line which does not vanish. Then a point R, 
and a scalar A, can be determined^ which shall satisfy 
the qnatemion equation, 

EA^.A,B,^..,-^EA^.A^^=^ A.AB; 
namely by assuming any origin Oj and writing, 
nir^ jr OA^.A,B,-{-,,.-^OA,.A^, 

^^- ^ A:Bri~TAjr^ ' 

z ^OA^^AiBi-k-.., 

(b,) For any assumed point C, let 

Qc = CA,.A^B,+ ...'^CA^.A^B^; 

then this quaternion sum may be transformed as 

Qc = Qh^CH.AB = {A + CH).AB; 

and therefore its tensor is, 

TQc = {A'+CH')^.AB, 

in which AB and CH denote lengths. 

(c) The least value of this tensor TQc is obtained by 
placing the point C at H; if then a quaternion be 
said to be a minimum when its tensor is such, we 
may write 

so that this minimum of Q^ is a vector. 

(rf.) The equation 

TQc = c = any scalar constant > TQh 


expresses that the locus of the variable point C is 
a spheric surface^ with its centre at the fixed point 
Hf and with a radius r, or CH, such that 

so that Hj as being thus the common centre of a 
series of concentric spheres, determined by the 
given system of right lines, may be said to be the 
Central Point, or simply the Centre, of that system. 

{e,) The equation 

TFQc = (?i = any scalar constant > TQh 
represents a right cylinder, of which the radius 
= (cj— A«.ZB»)* divided by ZB, and of which the 
axis of revolution is the line, 

FQc= QH^h.AB; 
wherefore this last right line, as being the common 
axis of a series of such right cylinders, may be 
called the Central Axis of the system. 

(/.) The equation 

SQc = ^a = any scalar constant 
^' represents a plane ; and all such planes are parallel 

to the Central Plane, of which the equation is 
SQc = 0. 

{g.) Prove that the central axis intersects the central plane 
perpendicularly, in the central point of the system. 

(A.) "When the n given vectors A^B^, ... A^B^ are parallel, 
and are therefore proportional to n scalars, ii, ... *«> 
the scalar h and the vector Qh vanish; and the 
centre ^is then determined by the equation, 

or by the expression, 

^^= *, + ... + *. ' 

where is again an arbitrary origin. 


14. (lUd. 1860.) 

(a.) The normal at the end of the variable vector p, to 
the surface of revolution of the sixth dimension^ 
which is represent^ by the equation 

{p*--a*y = 27a»(p-.a)*, (a) 

or by the system of the two equations^ 

p*^a* = 3<»aS {p-ay = ^»o», (a') 

and the tangent to the meridian at that pointy are 
respectively parallel to the two vectors, 

y = 2(p— a)— ^p, 

and T = 2(1— 2<)(p— a)4-^V; 

80 that they intersect the axis a, in points of which 

the vectors are, respectively, 


1^, and ?11Z:M!». 
l^t' (2-Q«-2 

(i.) If d^ be in the same meridian plane as p, then 

^(1— Q(4-^dp = Srrf^, and 5^ = —^. 

(c.) Under the same condition, 

4 =!<'-'>■ 

(rf.) The vector of the centre of curvature of the meridian^ 
at the end of the vector p, is, therefore, 

/o^^\""' 3 V 6 a— (4-Qp 

(tf.) The expressions in Example 38 give 

V* = a«^«(l-0% T« = a«^»(l-0'(4-0; 

{(t-pY = %«^«, and dp^ = |^^^'; 


the radius of curvature of the meridian is^ therefore^ 

and the length of an element of arc of that curve is 
ds =1 Tdp ^ ^Ta{-^^dt. 

(/.) The same expressions give 

thus the auxiliary scalar t is confined between the 
limits and 4, and we may write ^ = 2 vers ^, where 
^ is a real angle^ which varies continuously from 
to 2 IT ; the recent expression for the element of arc 
becomes^ therefore, 

ds =, 
and gives by integration 


if the arc s be measured from the point, say F, for 
which p = a, and which is common to all the meri- 
dians; and the total peripher}" of any one such 
curve is = 127rTa. 

(ff.) The value of <r gives 

4(<r^-a») = 3a«^(4-^), 16 (r(wr)» = -a*^» (4-^)^ ; 

if, then, we set aside the axis of revolution a, which 
is crossed by all the normals to the siuface (a), the 
siuface of centres of curvature which is touched by 
all those normals is represented by the equation, 

4(<r«— a»)» + 27a«(ra<r)« = (b) 

(ii.) The point F is common to the two surfaces (a) and 
(b), and is a singular point on each of them, being 
a triple point on (a), and a double point on (b); 
there is also at it an infinitely sharp cusp on (b), 


which tends to coincide with the axis a, but a 
determined tangent plane to (a)^ which is perpen- 
dicular to that axis, and to that cusp; and the 
pointy say F\ of which the vector = — a, is another 
and an exactly similar cusp on (b)^ but does not 
belong to (a). 

(i.) Besides the three universally eaincidefU intersections of 
the surface (a)^ with any transversal, drawn through 
its triple point F, in any given direction p, there are 
always tAree other real intersections y of which indeed 
one coincides with F if the transversal be perpen- 
dicular to the axis^ and for which the following is 
a general formula : 

(J.) The point, say F, of which the vector is p = 2o, is 
a double point of (a), near which that surface has a 
cusp, which coincides nearly with its tangent cone 
at that point; and the semi-angle of this cone 


(28p{p-a)=zaH^{Z+t), , 
1.2&X (p-a) = a»^»(3-4 

C25(p-a)r = aH'{\^t){^^^. 


>V.: i»*. ».♦•